How many questions from Permutations and Combinations appear in JEE?

Permutations and Combinations has appeared in 162 of the last 186 JEE Main and JEE Advanced papers — about 87% of them — contributing 220 questions in total across those papers.

Is Permutations and Combinations an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 87% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Permutations and Combinations questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Permutations and Combinations JEE Previous Year Questions with Answers

Q1·MathematicsIntegerJEE Advanced 2026

The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ______.

Q2·MathematicsSingle correctJEE Main 2026

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:

(A)18
(B)36
(C)39
(D)72

Q3·MathematicsSingle correctJEE Main 2026

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

(A)2670
(B)2840
(C)2920
(D)3600

Q4·MathematicsSingle correctJEE Main 2026

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the

10^{\text{th}}

floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

(A)2184
(B)3064
(C)7056
(D)11340

Q5·MathematicsNumericalJEE Main 2026

Two players

A

and

B

play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player

A

wins the series is __________.

Q6·MathematicsNumericalJEE Main 2026

Let

A = \{1, 2, 3, 4, 5, 6\}

. The number of one-one functions

f : A \to A

such that

f(1) \geq 3

,

f(3) \leq 4

and

f(2) + f(3) = 5

, is __________.

Q7·MathematicsSingle correctJEE Main 2026

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is :

(A)4100
(B)4140
(C)4230
(D)4290

Q8·MathematicsSingle correctJEE Main 2026

The number of ways of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

(A) $5040$
(B) $3050$
(C) $3410$
(D) $4320$

Q9·MathematicsSingle correctJEE Main 2026

The number of functions

f : \{1, 2, 3, 4\} \to \{a, b, c\}

, which are not onto, is:

(A) $48$
(B) $45$
(C) $51$
(D) $35$

Q10·MathematicsSingle correctJEE Main 2026

Let

A = \{(a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22\}

. Then

n(A)

is equal to:

(A)121
(B)124
(C)144
(D)169

Q11·MathematicsSingle correctJEE Main 2026

The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 5 and 7 such that each digit is used at least once, is :

(A)15400
(B)17800
(C)16800
(D)29400

Q12·MathematicsSingle correctJEE Main 2026

Let

p_n

denote the total number of triangles formed by joining the vertices of an

n

-side regular polygon. If

p_{n+1} - p_n = 66

, then the sum of all distinct prime divisors of

n

is:

(A) $7$
(B) $8$
(C) $5$
(D) $6$

Q13·MathematicsNumericalJEE Main 2026

Three persons enter in a lift at the ground floor. The lift will go upto

10^{th}

floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _______ .

Q14·MathematicsSingle correctJEE Main 2026

Let

S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}

. Let

x

be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let

y

be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

(A) $29x = 5y$
(B) $45x = 7y$
(C) $21x = 4y$
(D) $56x = 9y$

Q15·MathematicsSingle correctJEE Main 2026

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is :

(A)1580
(B)1578
(C)1579
(D)1581

Q16·MathematicsSingle correctJEE Main 2026

The largest value of n, for which

40^{n}

divides 60!, is

(A)13
(B)11
(C)12
(D)14

Q17·MathematicsNumericalJEE Main 2026

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ________

Q18·MathematicsSingle correctJEE Main 2026

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

(A)429
(B)384
(C)403
(D)455

Q19·MathematicsNumericalJEE Main 2026

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is _______ .

Q20·MathematicsNumericalJEE Main 2026

Let S denote the set of 4-digit numbers abcd such that a > b > c > d and P denote the set of 5-digit numbers having product of its digits equal to 20. Then n(S) + n(P) is equal to…………

Q21·MathematicsNumericalJEE Main 2026

Let ABC be a triangle. Consider four points

p_{1}

,

p_{2}

,

p_{3}

,

p_{4}

on the side AB, five points

p_{5}

,

p_{6}

,

p_{7}

,

p_{8}

,

p_{9}

on the side BC and four points

p_{10}

,

p_{11}

,

p_{12}

,

p_{13}

on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points

p_{1}

,

p_{2}

, ....

p_{13}

, is _________.

Q22·MathematicsNumericalJEE Main 2026

Let S be the set of the first 11 natural numbers. Then the number of elements in A = {B ⊆ S : n(B) ≥ 2 and the product of all elements of B is even} is ______ .

Q23·MathematicsSingle correctJEE Main 2026

The number of strictly increasing functions f from the set

\{1, 2, 3, 4, 5, 6\}

to the set

(1, 2, 3,…,9)

such that

f(i) \neq i

for

1 \leq i \leq 6

, is equal to :

(A)21
(B)27
(C)22
(D)28

Q24·MathematicsNumericalJEE Main 2026

Let S={(m, n): m, n ∈ {1, 2, 3, ....., 50}}. If the number of elements (m, n) in S such that

6^{m}+9^{n}

is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p + q is equal to……..

Q25·MathematicsSingle correctJEE Main 2026

The largest

n \in N

, for which

7^{n}

divides

101!

, is :

(A)16
(B)18
(C)15
(D)19

Q26·MathematicsNumericalJEE Advanced 2025

Let S be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in S, but 0210222 is NOT is S. Then the number of elements x in S such that at least one of the digits 0 and 1 appears exactly twice in x, is equal to _________ .

Q27·MathematicsSingle correctJEE Main 2025

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

(A) $230$
(B) $220$
(C) $200$
(D) $210$

Q28·MathematicsSingle correctJEE Main 2025

Let

p

be the number of all triangles that can be formed by joining the vertices of a regular polygon

P

of

n

sides and

q

be the number of all quadrilaterals that can be formed by joining the vertices of

P

. If

p+q=126

, then the eccentricity of the ellipse

\dfrac{x^2}{16}+\dfrac{y^2}{n}=1

is:

(A) $\dfrac{3}{4}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{\sqrt{7}}{4}$
(D) $\dfrac{1}{\sqrt{2}}$

Q29·MathematicsIntegerJEE Main 2025

For

n\ge2

, let

S_n

denote the set of all subsets of

\{1,2,\ldots,n\}

with no two consecutive numbers. For example

\{1,3,5\}\in S_5

, but

\{1,2,4\}\notin S_5

. Then

n(S_5)

is equal to ______.

Q30·MathematicsSingle correctJEE Main 2025

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

(A)165
(B)155
(C)145
(D)135

Q31·MathematicsIntegerJEE Main 2025

Let

m

and

n

,

(m<n)

be two 2-digit numbers. Then the total number of pairs

(m,n)

, such that

\gcd(m,n)=6

, is ______.

Q32·MathematicsSingle correctJEE Main 2025

Consider the sets

A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^2+y^2=25\}

,

B=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^2+9y^2=144\}

,

C=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:x^2+y^2\le4\}

, and

D=A\cap B

. The total number of one-one functions from the set

D

to the set

C

is:

(A)15120
(B)19320
(C)17160
(D)18290

Q33·MathematicsSingle correctJEE Main 2025

Line

L_1

of slope 2 and line

L_2

of slope

\dfrac{1}{2}

intersect at the origin

O

. In the first quadrant,

P_1,P_2,\ldots,P_{12}

are 12 points on line

L_1

and

Q_1,Q_2,\ldots,Q_9

are 9 points on line

L_2

. Then the total number of triangles, that can be formed having vertices at three of the 22 points

O,P_1,P_2,\ldots,P_{12},Q_1,Q_2,\ldots,Q_9

, is:

(A)1080
(B)1134
(C)1026
(D)1188

Q34·MathematicsIntegerJEE Main 2025

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number

n

be denoted by

W_n

. Let the probability

P(W_n)

of choosing the word

W_n

satisfy

P(W_n)=2P(W_{n-1})

,

n>1

. If

P(\text{CDBEA})=\dfrac{2^{\alpha}}{2^{\beta}-1}

,

\alpha,\beta\in\mathbb{N}

, then

\alpha+\beta

is equal to __________.

Q35·MathematicsIntegerJEE Main 2025

If the number of seven-digit numbers, such that the sum of their digits is even, is

m\cdot n\cdot 10^n

,

m,n\in\{1,2,3,\ldots,9\}

, then

m+n

is equal to __________.

Q36·MathematicsSingle correctJEE Main 2025

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure (shown in the figure) so that no row remains empty and at most one letter can be placed in a box, is:

(A)5880
(B)960
(C)840
(D)5760

Q37·MathematicsSingle correctJEE Main 2025

The largest

n\in N

such that

3^n

divides

50!

is:

(A)21
(B)22
(C)20
(D)23

Q38·MathematicsSingle correctJEE Main 2025

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to

(A)360
(B)45
(C)2520
(D)1820

Q39·MathematicsIntegerJEE Main 2025

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.

Q40·MathematicsSingle correctJEE Main 2025

If all the words with or without meaning made using all the letters of the word “KANPUR” are arranged as in a dictionary, then the word at

440^{\text{th}}

position in this arrangement, is:

(A)PRNAKU
(B)PRKANU
(C)PRKAUN
(D)PRNAUK

Q41·MathematicsSingle correctJEE Main 2025

Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P is:

(A)158
(B)173
(C)164
(D)161

Q42·MathematicsIntegerJEE Main 2025

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is ______.

Q43·MathematicsSingle correctJEE Main 2025

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

(A) $\frac{1}{4}$
(B) $\frac{2}{3}$
(C) $\frac{1}{3}$
(D) $\frac{1}{2}$

Q44·MathematicsSingle correctJEE Main 2025

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

(A)4608
(B)5720
(C)5719
(D)4607

Q45·MathematicsIntegerJEE Main 2025

Number of functions

f:\{1,2,\ldots,100\}\to\{0,1\}

, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________.

Q46·MathematicsIntegerJEE Main 2025

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is _______

Q47·MathematicsSingle correctJEE Main 2025

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:

(A)8575
(B)9100
(C)8925
(D)8750

Q48·MathematicsIntegerJEE Main 2025

Let

S=\{p_1,p_2,\ldots,p_{10}\}

be the set of first ten prime numbers. Let

A=S\cup P

, where P is the set of all possible products of distinct elements of S. Then the number of all ordered pairs

(x,y)

,

x\in S

,

y\in A

, such that x divides y, is _______

Q49·MathematicsIntegerJEE Main 2025

The number of ways,

5

boys and

4

girls can sit in a row so that either all the boys sit together or no two boys sit together, is __________.

Q50·MathematicsSingle correctJEE Main 2025

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

(A) $34000$
(B) $37000$
(C) $36000$
(D) $35000$

Q51·MathematicsSingle correctJEE Main 2025

From the English alphabet, five letters are chosen and arranged in alphabetical order. The number of ways in which the middle letter is "M" is:

(A)14950
(B)6084
(C)4356
(D)5148

Q52·MathematicsSingle correctJEE Main 2025

In a group of 3 girls and 4 boys, there are two boys

B_1

and

B_2

. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but

B_1

and

B_2

are not adjacent to each other, is:

(A)144
(B)72
(C)96
(D)120

Q53·MathematicsNumericalJEE Advanced 2024

Let

S = \{1, 2, 3, 4, 5, 6\}

and

X

be the set of all relations

R

from

S

to

S

that satisfy both the following properties: i.

R

has exactly 6 elements. ii. For each

(a, b) \in R

, we have

|a - b| \ge 2

. Let

Y = \{R \in X : \text{The range of } R \text{ has exactly one element}\}

and

Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}

. Let

n(A)

denote the number of elements in a set

A

. If

n(X) = {}^{m}C_6

, then the value of m is ______

Q54·MathematicsNumericalJEE Advanced 2024

Let

S = \{1, 2, 3, 4, 5, 6\}

and

X

be the set of all relations

R

from

S

to

S

that satisfy both the following properties: i.

R

has exactly 6 elements. ii. For each

(a, b) \in R

, we have

|a - b| \ge 2

. Let

Y = \{R \in X : \text{The range of } R \text{ has exactly one element}\}

and

Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}

. Let

n(A)

denote the number of elements in a set

A

. If the value of n(Y) + n(Z) is

k^2

, then

|k|

is ______

Q55·MathematicsIntegerJEE Advanced 2024

A group of 9 students

s_{1}, s_{2}, \ldots\ldots s_{9}

is to be divided to form three teams

X

,

Y

and

Z

of sizes 2, 3 and 4 respectively. Suppose that

s_{1}

cannot be selected for the team

X

, and

s_{2}

cannot be selected for team

Y

. Then the number of ways to form such teams, is ______ .

Q56·MathematicsNumericalJEE Main 2024

Let

A=\{(x, y):2x+3y=23,\ x, y\in\mathbb{N}\}

and

B=\{x:(x, y)\in A\}

. Then the number of one-one functions from A to B is equal to _______.

Q57·MathematicsNumericalJEE Main 2024

The number of integers, between 100 and 1000 having the sum of their digits equal to 14, is _______.

Q58·MathematicsSingle correctJEE Main 2024

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to

(A) $175$
(B) $181$
(C) $177$
(D) $179$

Q59·MathematicsNumericalJEE Main 2024

The number of

3

-digit numbers, formed using the digits

2,3,4,5

and

7

, when the repetition of digits is not allowed, and which are not divisible by

3

, is equal to ___

Q60·MathematicsSingle correctJEE Main 2024

Let

[t]

be the greatest integer less than or equal to

t

. Let

A

be the set of all prime factors of

2310

and

f:A\to\mathbb{Z}

be the function

f(x)=\left[\log_2\left(x^2+\left[\dfrac{x^3}{5}\right]\right)\right]

. The number of one-to-one functions from

A

to the range of

f

is:

(A) $20$
(B) $120$
(C) $25$
(D) $24$

Q61·MathematicsSingle correctJEE Main 2024

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at

315^{th}

position in this arrangement is:

(A)NRAGUP
(B)NRAGPU
(C)NRAPGU
(D)NRAPUG

Q62·MathematicsSingle correctJEE Main 2024

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

(A) $24$
(B) $56$
(C) $16$
(D) $48$

Q63·MathematicsSingle correctJEE Main 2024

Let

A = \{1, 3, 7, 9, 11\}

and

B = \{2, 4, 5, 7, 8, 10, 12\}

. Then the total number of one-one maps

f : A \to B

, such that

f(1) + f(3) = 14

, is:

(A) $180$
(B) $120$
(C) $480$
(D) $240$

Q64·MathematicsNumericalJEE Main 2024

The number of ways of getting a sum 16 on throwing a dice four times is ___.

Q65·MathematicsSingle correctJEE Main 2024

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is:

(A)OBBHJ
(B)HBBJO
(C)OBBJH
(D)JBBOH

Q66·MathematicsSingle correctJEE Main 2024

Let the set

S=\{2,4,8,16,\ldots,512\}

be partitioned into 3 sets

A,B,C

with an equal number of elements such that

A\cup B\cup C=S

and

A\cap B=B\cap C=A\cap C=\varnothing

. The maximum number of such possible partitions of

S

is equal to:

(A)1680
(B)1520
(C)1710
(D)1640

Q67·MathematicsNumericalJEE Main 2024

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is

Q68·MathematicsSingle correctJEE Main 2024

There are

5

points

P_1,P_2,P_3,P_4,P_5

on the side

AB

, excluding

A

and

B

, of a triangle

ABC

. Similarly there are

6

points

P_6,P_7,\ldots,P_{11}

on the side

BC

and

7

points

P_{12},P_{13},\ldots,P_{18}

on the side

CA

of the triangle. The number of triangles, that can be formed using the points

P_1,P_2,\ldots,P_{18}

as vertices, is:

(A) $776$
(B) $751$
(C) $796$
(D) $771$

Q69·MathematicsNumericalJEE Main 2024

The number of elements in the set

S=\{(x,y,z):x,y,z\in\mathbb{Z},\ x+2y+3z=42,\ x,y,z\ge0\}

equals ___

Q70·MathematicsSingle correctJEE Main 2024

If

n

is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then

n

is equal to:

(A) $47$
(B) $53$
(C) $51$
(D) $43$

Q71·MathematicsSingle correctJEE Main 2024

If for some

m

,

n

;

\ ^6C_m+2\left(^6C_{m+1}\right)+\ ^6C_{m+2}>\ ^8C_3

and

\ ^{n-1}P_3:\ ^nP_4=1:8

, then

\ ^nP_{m+1}+\ ^{n+1}C_m

is equal to

(A)380
(B)376
(C)384
(D)372

Q72·MathematicsSingle correctJEE Main 2024

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

(A)406
(B)130
(C)142
(D)136

Q73·MathematicsNumericalJEE Main 2024

The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______.

Q74·MathematicsNumericalJEE Main 2024

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is ______.

Q75·MathematicsSingle correctJEE Main 2024

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to:

(A)18
(B)16
(C)12
(D)15

Q76·MathematicsNumericalJEE Main 2024

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is ______.

Q77·MathematicsSingle correctJEE Main 2024

Let

\alpha=\dfrac{(4!)!}{(4!)^{3!}}

and

\beta=\dfrac{(5!)!}{(5!)^{4!}}

. Then :

(A) $\alpha\in\mathbb{N}$ and $\beta\notin\mathbb{N}$
(B) $\alpha\notin\mathbb{N}$ and $\beta\in\mathbb{N}$
(C) $\alpha\in\mathbb{N}$ and $\beta\in\mathbb{N}$
(D) $\alpha\notin\mathbb{N}$ and $\beta\notin\mathbb{N}$

Q78·MathematicsSingle correctJEE Main 2024

^{n-1}C_r=(k^2-8)\cdot{}^{n}C_{r+1}

if and only if:

(A) $2\sqrt2<k\le 3$
(B) $2\sqrt3<k\le 3\sqrt2$
(C) $2\sqrt3<k<3\sqrt3$
(D) $2\sqrt2<k<2\sqrt3$

Q79·MathematicsIntegerJEE Advanced 2023

Let

X

be the set of all five digit numbers formed using 1, 2, 2, 2, 4, 4, 0. For example, 22240 is in

X

while 02244 and 44422 are not in

X

. Suppose that each element of

X

has an equal chance of being chosen. Let

p

be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of

38p

is equal to

Q80·MathematicsSingle correctJEE Main 2023

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is:

(A)22
(B)18
(C)21
(D)20

Q81·MathematicsNumericalJEE Main 2023

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is____

Q82·MathematicsNumericalJEE Main 2023

The number of elements in the set

\{n \in \mathbb{N} : 10 \le n \le 100 \text{ and } 3^n - 3 \text{ is a multiple of } 7\}

is____

Q83·MathematicsNumericalJEE Main 2023

Total number of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is

Q84·MathematicsNumericalJEE Main 2023

The number of seven digit positive integers formed using the digits

1,\,2,\,3

and

4

only and sum of the digits equal to

12

is _____.

Q85·MathematicsSingle correctJEE Main 2023

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is

(A)327
(B)326
(C)328
(D)324

Q86·MathematicsNumericalJEE Main 2023

Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Q87·MathematicsSingle correctJEE Main 2023

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to

(A)120
(B)132
(C)72
(D)96

Q88·MathematicsSingle correctJEE Main 2023

If the letters of the word MATHS are permuted and all possible words so formed are arranged in a dictionary with serial numbers, then the serial number of the word THAMS is

(A) $103$
(B) $104$
(C) $101$
(D) $102$

Q89·MathematicsSingle correctJEE Main 2023

The number of triplets

(x, y, z)

, where

x, y, z

are distinct non negative integers satisfying

x + y + z = 15

, is

(A)80
(B)114
(C)92
(D)136

Q90·MathematicsNumericalJEE Main 2023

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _______ .

Q91·MathematicsNumericalJEE Main 2023

The sum of all the four-digit numbers that can be formed using all the digits

2,1,2,3

is equal to _______ .

Q92·MathematicsNumericalJEE Main 2023

The number of permutations, of the digits

1,2,3,\ldots,7

without repetition, which neither contain the string 153 nor the string 2467, is _________.

Q93·MathematicsSingle correctJEE Main 2023

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is:

(A) $3360$
(B) $1680$
(C) $560$
(D) $1120$

Q94·MathematicsNumericalJEE Main 2023

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is _________.

Q95·MathematicsNumericalJEE Main 2023

Let

R=\{a,b,c,d,e\}

and

S=\{1,2,3,4\}

. Total number of onto functions

f:R\to S

such that

f(a)\neq1

, is equal to

Q96·MathematicsSingle correctJEE Main 2023

The number of arrangements of the letter of the word "INDEPENDENCE" in which all the vowels always occur together is

(A)16800
(B)14800
(C)18000
(D)33600

Q97·MathematicsNumericalJEE Main 2023

The largest natural number

n

such that

3^{n}

divides

66!

is

Q98·MathematicsSingle correctJEE Main 2023

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is

(A) $126(5!)^{2}$
(B) $7(360)^{2}$
(C)720
(D) $7(720)^{2}$

Q99·MathematicsSingle correctJEE Main 2023

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is

(6!)k

, then k is equal to

(A)1890
(B)945
(C)2835
(D)5670

Q100·MathematicsSingle correctJEE Main 2023

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

(A)580
(B)582
(C)578
(D)576

Q101·MathematicsNumericalJEE Main 2023

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is

Q102·MathematicsNumericalJEE Main 2023

The number of ways of giving

20

distinct oranges to

3

children such that each child gets at least one orange is _____.

Q103·MathematicsNumericalJEE Main 2023

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _______ .

Q104·MathematicsNumericalJEE Main 2023

The number of

3

-digit numbers, that are divisible by either

2

or

3

but not divisible by

7

, is _______ .

Q105·MathematicsNumericalJEE Main 2023

Number of integral solutions to the equation

x+y+z=21

, where

x\geq 1

,

y\geq 3

,

z\geq 4

, is equal to

Q106·MathematicsNumericalJEE Main 2023

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is

Q107·MathematicsNumericalJEE Main 2023

If

^{2n+1}P_{n-1}:{}^{2n-1}P_{n}=11:21

, then

n^2+n+15

is equal to:

Q108·MathematicsNumericalJEE Main 2023

Let

A=[a_{ij}],\ a_{ij}\in\mathbb{Z}\cap[0,4],\ 1\le i,j\le2

. The number of matrices

A

such that the sum of all entries is a prime number

p\in(2,13)

is

Q109·MathematicsNumericalJEE Main 2023

Let 5 digit numbers be constructed using the digits

0, 2, 3, 4, 7, 9

with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is

Q110·MathematicsNumericalJEE Main 2023

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to

Q111·MathematicsNumericalJEE Main 2023

Let

S=\{1,2,3,4,5,6\}

. Then the number of one-one functions

f:S\to P(S)

, where

P(S)

denote the power set of S, such that

f(n)\subset f(m)

where

n<m

is

Q112·MathematicsNumericalJEE Main 2023

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits

1,2,3

and

5

, and are divisible by

15

, is equal to

Q113·MathematicsSingle correctJEE Main 2023

The number of ways of selecting two numbers

a

and

b

,

a \in \{2, 4, 6, \ldots, 100\}

and

b \in \{1, 3, 5, \ldots, 99\}

, such that

2

is the remainder when

a + b

is divided by

23

, is:

(A) $268$
(B) $108$
(C) $54$
(D) $186$

Q114·MathematicsNumericalJEE Main 2023

The number of seven-digit odd numbers that can be formed using all the seven digits

1, 2, 2, 2, 3, 3, 5

is _______.

Q115·MathematicsSingle correctJEE Main 2023

The number of 3 digit numbers, that are divisible by either

3

or

4

but not divisible by

48

, is:

(A) $507$
(B) $432$
(C) $472$
(D) $400$

Q116·MathematicsSingle correctJEE Main 2023

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

(A) $84$
(B) $79$
(C) $89$
(D) $86$

Q117·MathematicsNumericalJEE Main 2023

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ________ .

Q118·MathematicsNumericalJEE Main 2023

The total number of 4-digit numbers whose greatest common divisor with

54

is

2

, is _____.

Q119·MathematicsNumericalJEE Main 2023

If all the six digit numbers

x_1 x_2 x_3 x_4 x_5 x_6

with

0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6

are arranged in the increasing order, then the sum of the digits in the 72th number is ________ .

Q120·MathematicsNumericalJEE Main 2023

Suppose Anil's mother wants to give

5

whole fruits to Anil from a basket of

7

red apples,

5

white apples and

8

oranges. If in the selected

5

fruits, at least

2

oranges, at least one red apple and at least one white apple must be given, then the number of ways Anil's mother can offer

5

fruits to Anil is

Q121·MathematicsNumericalJEE Main 2023

Let

x

and

y

be distinct integers where

1\le x\le 25

and

1\le y\le 25

. Then, the number of ways of choosing

x

and

y

, such that

x+y

is divisible by

5

, is _______.

Q122·MathematicsNumericalJEE Main 2023

Let

S=\{1,2,3,5,7,10,11\}

. The number of non-empty subsets of

S

that have the sum of all elements a multiple of

3

, is _______.

Q123·MathematicsSingle correctJEE Main 2023

The number of numbers, strictly between

5000

and

10000

that can be formed using the digits

1,3,5,7,9

without repetition, is

(A) $12$
(B) $120$
(C) $72$
(D) $6$

Q124·MathematicsSingle correctJEE Main 2023

The number of square matrices of order

5

with entries from the set

\{0,1\}

, such that the sum of all the elements in each row is

1

and the sum of all the elements in each column is also

1

, is

(A) $125$
(B) $225$
(C) $150$
(D) $120$

Q125·MathematicsSingle correctJEE Main 2023

The number of integers, greater than

7000

that can be formed, using the digits

3,5,6,7,8

without repetition, is

(A) $168$
(B) $220$
(C) $120$
(D) $48$

Q126·MathematicsNumericalJEE Main 2023

A boy needs to select five courses from

12

available courses, out of which

5

courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is _______ .

Q127·MathematicsNumericalJEE Main 2023

The number of

9

digit numbers, that can be formed using all the digits of the number

123412341

so that the even digits occupy only even places, is _______ .

Q128·MathematicsNumericalJEE Advanced 2022

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is __________.

Q129·MathematicsSingle correctJEE Advanced 2022

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

(A)21816
(B)85536
(C)12096
(D)156816

Q130·MathematicsNumericalJEE Main 2022

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is____,

Q131·MathematicsNumericalJEE Main 2022

The number of matrices of order

3 \times 3

, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is__________.

Q132·MathematicsNumericalJEE Main 2022

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then

b + 3g

is equal to

Q133·MathematicsNumericalJEE Main 2022

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is

\alpha \times 5^{6}

, then

\alpha

is equal to ______.

Q134·MathematicsNumericalJEE Main 2022

The number of functions f, from the set

A = \left\{x \in N : x^{2} - 10x + 9 \le 0\right\}

to the set

B = \left\{n^{2} : n \in N\right\}

such that

f(x) \le (x-3)^{2} + 1

, for every

x \in A

, is __________.

Q135·MathematicsNumericalJEE Main 2022

The number of 5-digit natural numbers, such that the product of their digits is 36, is

Q136·MathematicsNumericalJEE Main 2022

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1,2,3,4,5 and 6 without repetition of digits. Then the total number of such numbers is ______________.

Q137·MathematicsNumericalJEE Main 2022

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ________.

Q138·MathematicsSingle correctJEE Main 2022

The total number of functions, f :

\{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}

such that f(1) + f(2) = f(3), is equal to :

(A)60
(B)90
(C)108
(D)126

Q139·MathematicsNumericalJEE Main 2022

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to______.

Q140·MathematicsNumericalJEE Main 2022

Let

b_1b_2b_3b_4

be a 4-element permutation with

b_i \in \{1, 2, 3, \ldots, 100\}

for

1 \leq i \leq 4

and

b_i \neq b_j

for

i \neq j

, such that either

b_1, b_2, b_3

are consecutive integers or

b_2, b_3, b_4

are consecutive integers. Then the number of such permutations

b_1b_2b_3b_4

is equal to ________.

Q141·MathematicsSingle correctJEE Main 2022

The number of ways to distribute 30 identical candies among four children

C_1

,

C_2

,

C_3

and

C_4

so that

C_2

receives atleast 4 and atmost 7 candies,

C_3

receives atleast 2 and atmost 6 candies, is equal to

(A)205
(B)615
(C)510
(D)430

Q142·MathematicsSingle correctJEE Main 2022

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is

(A)36
(B)48
(C)60
(D)72

Q143·MathematicsNumericalJEE Main 2022

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ________.

Q144·MathematicsNumericalJEE Main 2022

Let A be a matrix of order

2\times 2

, whose entries are from the set

\{0, 1, 2, 3, 4, 5\}

. If the sum of all the entries of A is a prime number p,

2<p<8

, then the number of such matrices A is :

Q145·MathematicsNumericalJEE Main 2022

There are ten boys

B_{1}

,

B_{2}

, ....,

B_{10}

and five girls

G_{1}

,

G_{2}

, ....,

G_{5}

in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both

B_{1}

and

B_{2}

together should not be the members of a group, is____________.

Q146·MathematicsNumericalJEE Main 2022

The total number of 3−digit numbers, whose greatest common divisor with 36 is 2, is ______ .

Q147·MathematicsNumericalJEE Main 2022

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is ______.

Q148·MathematicsNumericalJEE Main 2022

The total number of three-digit numbers, with one digit repeated exactly two times, is

Q149·MathematicsNumericalJEE Main 2022

Let A be a

3 \times 3

matrix having entries from. the set {−1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ________

Q150·MathematicsNumericalJEE Main 2022

The number of one-one function

f:\{a,b,c,d\}\to\{0,1,2,\ldots,10\}

such that

2f(a)-f(b)+3f(c)+f(d)=0

is ______ .

Q151·MathematicsNumericalJEE Main 2022

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

Q152·MathematicsMultiple correctJEE Advanced 2021

Let

S_1 = \{(i, j, k) : i, j, k \in \{1, 2, \ldots, 10\}\}

S_2 = \{(i, j) : 1 \le i < j + 2 \le 10,\ i, j \in \{1, 2, \ldots, 10\}\}

,

S_3 = \{(i, j, k, l) : 1 \le i < j < k < l,\ i, j, k, l \in \{1, 2, \ldots, 10\}\}

. and

S_4 = \{(i, j, k, l) : i, j, k

and

l

are distinct elements in

\{1, 2, \ldots, 10\}\}

. If the total number of elements in the set

S_r

is

n_r

, r = 1, 2, 3, 4, then which of the following statements is (are) TRUE?

(A) $n_1 = 1000$
(B) $n_2 = 44$
(C) $n_3 = 220$
(D) $\frac{n_4}{12} = 420$

Q153·MathematicsNumericalJEE Main 2021

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ______ .

Q154·MathematicsSingle correctJEE Main 2021

Let

P_{1}

,

P_{2}

, .....,

P_{15}

be 15 points on a circle. The number of distinct triangles formed by points

P_{i}, P_{j}, P_{k}

such that i + j + k ≠ 15, is :

(A)12
(B)419
(C)443
(D)455

Q155·MathematicsNumericalJEE Main 2021

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ______.

Q156·MathematicsNumericalJEE Main 2021

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ______.

Q157·MathematicsNumericalJEE Main 2021

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is __________.

Q158·MathematicsNumericalJEE Main 2021

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A ⊆ S : A ≠ φ and the sum of all the elements of A is not a multiple of 3} is ___________ .

Q159·MathematicsNumericalJEE Main 2021

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.

Q160·MathematicsNumericalJEE Main 2021

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ________.

Q161·MathematicsNumericalJEE Main 2021

If

^{1}P_{1} + 2 \cdot {}^{2}P_{2} + 3 \cdot {}^{3}P_{3} + ... + 15 \cdot {}^{15}P_{15} = {}^{q}P_{r} - s

,

0 \leq s \leq 1

, then

^{q+s}C_{r-s}

is equal to ________.

Q162·MathematicsNumericalJEE Main 2021

Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number

(10)^{10}

.

(11)^{10}

.

(13)^{13}

is equal to……….

Q163·MathematicsSingle correctJEE Main 2021

If

{}^n P_r = {}^n P_{r+1}

and

{}^n C_r = {}^n C_{r-1}

, then the value of r is equal to :

(A)4
(B)1
(C)3
(D)2

Q164·MathematicsNumericalJEE Main 2021

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is

100k

, then

k

is equal to.......

Q165·MathematicsNumericalJEE Main 2021

If the digits are not allowed to repeat in any number formed by using the digits 0,2,4,6,8, then the number of all numbers greater than 10,000 is equal to........

Q166·MathematicsNumericalJEE Main 2021

Let

A = \{0,1,2,3,4,5,6,7\}

. Then the number of bijective functions

f : A \to A

such that

f(1) + f(2) = 3 - f(3)

is equal to...............

Q167·MathematicsNumericalJEE Main 2021

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ..........

Q168·MathematicsSingle correctJEE Main 2021

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is:

(A) $26664$
(B) $122664$
(C) $122234$
(D) $22264$

Q169·MathematicsNumericalJEE Main 2021

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

Q170·MathematicsSingle correctJEE Main 2021

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :

(A) $364$
(B) $240$
(C) $333$
(D) $360$

Q171·MathematicsSingle correctJEE Main 2021

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

(A)5
(B)2
(C)4
(D)6

Q172·MathematicsSingle correctJEE Main 2021

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let

\alpha

be the number of triangles having these points from different sides as vertices and

\beta

be the number of quadrilaterals having these points from different sides as vertices. Then

(\beta-\alpha)

is equal to :

(A) $795$
(B) $1173$
(C) $1890$
(D) $717$

Q173·MathematicsSingle correctJEE Main 2021

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only is

(A)77
(B)42
(C)35
(D)82

Q174·MathematicsSingle correctJEE Main 2021

A natural number has prime factorization given by

n = 2^{x} 3^{y} 5^{z}

, where y and z are such that

y + z = 5

and

y^{-1} + z^{-1} = \frac{5}{6}

,

y > z

. Then the number of odd divisors of n, including 1, is :

(A)11
(B)6x
(C)12
(D)6

Q175·MathematicsNumericalJEE Main 2021

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is

Q176·MathematicsSingle correctJEE Main 2021

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A × B. Then:

(A)y = 273x
(B)2y = 91x
(C)y = 91x
(D)2y = 273x

Q177·MathematicsSingle correctJEE Main 2021

The total number of positive integral solutions (x, y, z) such that

xyz = 24

is

(A) $36$
(B) $45$
(C) $24$
(D) $30$

Q178·MathematicsNumericalJEE Main 2021

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is _________.

Q179·MathematicsNumericalJEE Main 2021

Let M be any

3 \times 3

matrix with entries from the set

\{0, 1, 2\}

. The maximum number of such matrices, for which the sum of diagonal elements of

M^T M

is seven, is __

Q180·MathematicsNumericalJEE Main 2021

The students

S_{1},S_{2},\ldots,S_{10}

are to be divided into 3 groups A, B and

C

such that each group has at least one student and the group

C

has at most 3 students. Then the total number of possibilities of forming such groups is________.

Q181·MathematicsSingle correctJEE Main 2021

A scientific committee is to formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed is:

(A)560
(B)1050
(C)1625
(D)575

Q182·MathematicsNumericalJEE Advanced 2020

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is ________

Q183·MathematicsNumericalJEE Advanced 2020

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ________

Q184·MathematicsSingle correctJEE Main 2020

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

(A) $2!3!4!$
(B) $(3!)^{3}.(4!)$
(C) $(3!)^{2}.(4!)$
(D) $3!(4!)^{3}$

Q185·MathematicsNumericalJEE Main 2020

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is __________.

Q186·MathematicsNumericalJEE Main 2020

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ‘SYLLABUS’ such that two letters are distinct and two letters are alike, is ____.

Q187·MathematicsSingle correctJEE Main 2020

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

(A)2255
(B)3000
(C)2250
(D)1500

Q188·MathematicsNumericalJEE Main 2020

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is ……

Q189·MathematicsSingle correctJEE Main 2020

The value of

(2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \ldots

up to

51^{th}

term

) + (1! - 2! + 3! - \ldots

upt to

51^{th}

term

)

is equal to:

(A)1
(B) $1 + (51)!$
(C) $1 + (52)!$
(D) $1 - 51(51)!$

Q190·MathematicsNumericalJEE Main 2020

The total number of 3

-

digit numbers, whose sum of digits is 10, is ____________.

Q191·MathematicsSingle correctJEE Main 2020

If the number of five digit numbers with distinct digits and 2 at the

10^{th}

place is 336 k, then k is equal to:

(A)6
(B)7
(C)4
(D)8

Q192·MathematicsNumericalJEE Main 2020

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that the most three of them are red is

Q193·MathematicsNumericalJEE Main 2020

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ___________.

Q194·MathematicsSingle correctJEE Main 2020

The number of ordered pairs (r, k) for which

6\cdot\,^{35}C_r=(k^{2}-3)\cdot\,^{36}C_{r+1}

, where k is an integer, is:

(A)3
(B)6
(C)4
(D)2

Q195·MathematicsSingle correctJEE Main 2020

Total number of 6 - digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is

(A) $5^{6}$
(B) $6!$
(C) $\frac{5}{2}(6!)$
(D) $\frac{1}{2}(6!)$

Q196·MathematicsNumericalJEE Advanced 2019

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ______

Q197·MathematicsSingle correctJEE Main 2019

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to:

(A)24
(B)28
(C)27
(D)25

Q198·MathematicsSingle correctJEE Main 2019

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :

(A) $2^{20}$
(B) $2^{20}+1$
(C) $2^{21}$
(D) $2^{20}-1$

Q199·MathematicsSingle correctJEE Main 2019

The number of 6 digit numbers hat can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digits is repeated, is

(A)36
(B)60
(C)72
(D)48

Q200·MathematicsSingle correctJEE Main 2019

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is

(A)210
(B)180
(C)170
(D)190

Q201·MathematicsSingle correctJEE Main 2019

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then:

(A) $n=m-8$
(B) $m+n=68$
(C) $m=n=78$
(D) $m=n=68$

Q202·MathematicsSingle correctJEE Main 2019

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is:

(A) $180$
(B) $175$
(C) $162$
(D) $160$

Q203·MathematicsSingle correctJEE Main 2019

The number of four \u2013 digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is:

(A)360
(B)288
(C)310
(D)306

Q204·MathematicsSingle correctJEE Main 2019

There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is :

(A) $12$
(B) $11$
(C) $9$
(D) $7$

Q205·MathematicsSingle correctJEE Main 2019

Consider three boxes, each containing 10 balls labelled 1, 2, ....., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by

n_i

, the label of the ball drawn from the

i^{th}

box,

(i = 1, 2, 3)

. Then, the number of ways in which the balls can be chosen such that

n_1 < n_2 < n_3

is:

(A)120
(B)82
(C)240
(D)164

Q206·MathematicsSingle correctJEE Main 2019

Let

S=\{1, 2, 3, ....., 100\}

. The number of non-empty subsets A of S such that the product of elements in A is even is:

(A) $2^{100}-1$
(B) $2^{50}(2^{50}-1)$
(C) $2^{50}-1$
(D) $2^{50}+1$

Q207·MathematicsSingle correctJEE Main 2019

Let S be the set of all triangle in the xy

-

plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is:

(A)9
(B)18
(C)32
(D)36

Q208·MathematicsSingle correctJEE Main 2019

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is:

(A)500
(B)200
(C)300
(D)350

Q209·MathematicsSingle correctJEE Main 2019

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to:

(A)250
(B)374
(C)372
(D)375

Q210·MathematicsNumericalJEE Advanced 2018

Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If

\alpha

is the number of one-one functions from X to Y and

\beta

is the number of onto functions from Y to X, then the value of

\frac{1}{5!}(\beta - \alpha)

is ______ .

Q211·MathematicsSingle correctJEE Advanced 2018

In a high school, a committee has to be formed from a group of 6 boys

M_{1}

,

M_{2}

,

M_{3}

,

M_{4}

,

M_{5}

,

M_{6}

and 5 girls

G_{1}

,

G_{2}

,

G_{3}

,

G_{4}

,

G_{5}

. (i) Let

\alpha_{1}

be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let

\alpha_{2}

be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let

\alpha_{3}

be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls. (iv) Let

\alpha_{4}

be the total number of ways in which the committee can be formed such that the committee has 4 members, having atleast 2 girls and such that both

M_{1}

and

G_{1}

are NOT in the committee together. The correct option is :

LIST-I	LIST-II
P.The value of $\alpha_{1}$ is	1.136
Q.The value of $\alpha_{2}$ is	2.189
R.The value of $\alpha_{3}$ is	3.192
S.The value of $\alpha_{4}$ is	4.200
	5.381
	6.461

(A)P $\to$ 4; Q $\to$ 6; R $\to$ 2; S $\to$ 1
(B)P $\to$ 1; Q $\to$ 4; R $\to$ 2; S $\to$ 3
(C)P $\to$ 4; Q $\to$ 6; R $\to$ 5; S $\to$ 2
(D)P $\to$ 4; Q $\to$ 2; R $\to$ 3; S $\to$ 1

Q212·MathematicsNumericalJEE Advanced 2018

The number of 5 digit numbers which are divisible by 4, with digits from the set

\{1, 2, 3, 4, 5\}

and the repetition of digits is allowed, is ______ .

Q213·MathematicsSingle correctJEE Advanced 2017

Let

S = \{1, 2, 3, \dots, 9\}

. For

k = 1, 2, \dots, 5

, let

N_{k}

be the number of subsets of

S

, each containing five elements out of which exactly

k

are odd. Then

N_{1} + N_{2} + N_{3} + N_{4} + N_{5} =

(A) $210$
(B) $252$
(C) $125$
(D) $126$

Q214·MathematicsSingle correctJEE Advanced 2017

How many

3 \times 3

matrices

M

with entries from

\{0, 1, 2\}

are there, for which the sum of the diagonal entries of

M^{T}M

is

5

?

(A) $126$
(B) $198$
(C) $162$
(D) $135$

Q215·MathematicsIntegerJEE Advanced 2017

Words of length 10 are formed using the letters, A, B, C, D, E, F, G, H, I, J. Let

x

be the number of such words where no letter is repeated ; and let

y

be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then,

\frac{y}{9x} =

Q216·MathematicsSingle correctJEE Advanced 2016

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

(A)380
(B)320
(C)260
(D)95

Q217·MathematicsIntegerJEE Advanced 2015

Let

n

be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let

m

be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of

\dfrac{m}{n}

is

Q218·MathematicsIntegerJEE Advanced 2014

Let

n_1 < n_2 < n_3 < n_4 < n_5

be positive integers such that

n_1 + n_2 + n_3 + n_4 + n_5 = 20

. Then the number of such distinct arrangements

(n_1, n_2, n_3, n_4, n_5)

is __________

Q219·MathematicsIntegerJEE Advanced 2014

Let

n \geq 2

be an integer. Take

n

distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of

n

is __________

Q220·MathematicsSingle correctJEE Advanced 2014

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

(A)264
(B)265
(C)53
(D)67

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