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Permutation and Combination — JEE Main & Advanced Mathematics PYQs

503 previous year questions from Permutation and Combination with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

503 questionsMathematicsSolutions on every page
1

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 _____________.

JEE Advanced 2026 Solution View → 2

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let X be the absolute value of the differen

JEE Advanced 2026 Solution View → 3

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:

JEE Main 2026 Solution View → 4

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^ th floor. The number of ways, in which any 4 persons e

JEE Main 2026 Solution View → 5

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 re

JEE Main 2026 Solution View → 6

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 :

JEE Main 2026 Solution View → 7

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 pla

JEE Main 2026 Solution View → 8

Let A= (a,b,c): a,b,c are non-negative integers and a+b+2c=22 . Then n(A) is equal to:

JEE Main 2026 Solution View → 9

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

JEE Main 2026 Solution View → 10

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 i

JEE Main 2026 Solution View → 11

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 :

JEE Main 2026 Solution View → 12

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, i

JEE Main 2026 Solution View → 13

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

JEE Main 2026 Solution View → 14

The largest value of n , for which 40^ n divides 60! , is

JEE Main 2026 Solution View → 15

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

JEE Main 2026 Solution View → 16

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 _ _ _ _

JEE Main 2026 Solution View → 17

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

JEE Main 2026 Solution View → 18

Let S denote the set of 4-digit numbers a b c d 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

JEE Main 2026 Solution View → 19

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

JEE Main 2026 Solution View → 20

Let ABC be a triangle. Consider four points p ₁, p ₂, p ₃, p ₄ on the side AB, five points p₅, p₆, p₇, p₈, p₉ on the side B C , and four points p₁₀, p₁₁, p₁₂, p₁₃ on the side AC. N

JEE Main 2026 Solution View → 21

The largest n N , for which 7^ n divides 101!, is :

JEE Main 2026 Solution View → 22

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) i for 1 i 6 , is equal to :

JEE Main 2026 Solution View → 23

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+

JEE Main 2026 Solution View → 24

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 in S . Then the number of elements x i

JEE Advanced 2025 Solution View → 25

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

JEE Main 2025 Solution View → 26

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 jo

JEE Main 2025 Solution View → 27

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 ca

JEE Main 2025 Solution View → 28

For n 2 , let S_n denote the set of all subsets of 1,2 . . ., n with no two consecutive numbers. For example 1,3,5 S ₆ , but 1,2,4 S ₆ . Then n ( ~S ₅ ) is equal to ________

JEE Main 2025 Solution View → 29

Let m and n ,( m n ) be two 2-digit numbers. Then the total numbers of pairs (m, n) , such that gcd (m, n)=6 , is ________

JEE Main 2025 Solution View → 30

Line L₁ of slope 2 and line L₂ of slope 1 2 intersect at the origin O . In the first quadrant, P ₁, P ₂, . P ₁₂ are 12 points on line L₁ and Q₁, Q₂, . . Q₉ are 9 points on line L₂

JEE Main 2025 Solution View → 31

If the number of seven-digit numbers, such that the sum of their digits is even, is m n 10^ n ; m, n 1,2,3, , 9 , then m+n is equal to _______

JEE Main 2025 Solution View → 32

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

JEE Main 2025 Solution View → 33

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

JEE Main 2025 Solution View → 34

The largest n N such that 3^ n divides 50 ! is:

JEE Main 2025 Solution View → 35

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^ th position in this arrangement, is

JEE Main 2025 Solution View → 36

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

JEE Main 2025 Solution View → 37

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 twi

JEE Main 2025 Solution View → 38

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

JEE Main 2025 Solution View → 39

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 mo

JEE Main 2025 Solution View → 40

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 g

JEE Main 2025 Solution View → 41

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 -

JEE Main 2025 Solution View → 42

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

JEE Main 2025 Solution View → 43

In a group of 3 girls and 4 boys, there are two boys B₁ and B₂ . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all

JEE Main 2025 Solution View → 44

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

JEE Main 2025 Solution View → 45

A group of 9 students, s ₁, ~s ₂, , s ₉ , is to be divided to from three teams X , Y , and Z of sizes 2,3 , and 4 , respectively. Suppose that s₁ cannot be selected for the team X

JEE Advanced 2024 Solution View → 46

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

JEE Main 2024 Solution View → 47

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 :

JEE Main 2024 Solution View → 48

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__________

JEE Main 2024 Solution View → 49

Let 0 r n . If ^ n +1 C _ r +1 : ^n C _ r : ^ n -1 C _ r -1 =55: 35: 21 , then 2 n +5 r is equal to:

JEE Main 2024 Solution View → 50

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 :

JEE Main 2024 Solution View → 51

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

JEE Main 2024 Solution View → 52

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 50^ th word is :

JEE Main 2024 Solution View → 53

The number of ways of getting a sum 16 on throwing a dice four times is______

JEE Main 2024 Solution View → 54

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 _____

JEE Main 2024 Solution View → 55

There are 5 points P₁, P₂, P₃, P₄, P₅ on the side A B , excluding A and B , of a triangle A B C . Similarly there are 6 points P ₆, P ₇, , P ₁₁ on the side BC and 7 points P ₁₂, P

JEE Main 2024 Solution View → 56

The lines L 1 , L 2 , . .. , L 20 are distinct. For n = 1 , 2 , 3 , . .. , 10 all the lines L 2 n − 1 are parallel to each other and all the lines L 2 n pass through a given point

JEE Main 2024 Solution View → 57

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:

JEE Main 2024 Solution View → 58

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

JEE Main 2024 Solution View → 59

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 ______.

JEE Main 2024 Solution View → 60

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

JEE Main 2024 Solution View → 61

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

JEE Main 2024 Solution View → 62

All the letters of the word G T W E N T Y 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 G T

JEE Main 2024 Solution View → 63

Let α = ( 4 ! ) ! ( 4 ! ) 3 ! and β = ( 5 ! ) ! ( 5 ! ) 4 ! . Then :

JEE Main 2024 Solution View → 64

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 h

JEE Advanced 2023 Solution View → 65

Consider the 6 × 6 square in the figure. Let A 1 , A 2 , … , A 49 be the points of intersections (dots in the picture) in some order. We say that A i and A j are friends

JEE Advanced 2023 Solution View → 66

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

JEE Main 2023 Solution View → 67

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

JEE Main 2023 Solution View → 68

All words, with or without meaning, are made using all the letters of the word M O N D A Y . These words are written as in a dictionary with serial numbers. The serial number of th

JEE Main 2023 Solution View → 69

Total numbers 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 ________

JEE Main 2023 Solution View → 70

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 _ _ _ _ _ _ _ .

JEE Main 2023 Solution View → 71

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

JEE Main 2023 Solution View → 72

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 leas

JEE Main 2023 Solution View → 73

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

JEE Main 2023 Solution View → 74

The number of triplets x , y , z where x , y , z are distinct non negative integers satisfying x + y + z = 15 , is

JEE Main 2023 Solution View → 75

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

JEE Main 2023 Solution View → 76

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 the

JEE Main 2023 Solution View → 77

The sum of all the four-digit numbers that can be formed using all the digits 2 ,   1 ,   2 ,   3 is equal to ____.

JEE Main 2023 Solution View → 78

The number of permutations, of the digits 1 , 2 , 3 , … , 7 without repetition, which neither contain the string 153 nor the string 2467 , is _______ .

JEE Main 2023 Solution View → 79

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 numbers of persons,

JEE Main 2023 Solution View → 80

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

JEE Main 2023 Solution View → 81

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

JEE Main 2023 Solution View → 82

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

JEE Main 2023 Solution View → 83

The largest natural number n such that 3 n divides 66 ! is _ _ _ _ _ _ _

JEE Main 2023 Solution View → 84

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

JEE Main 2023 Solution View → 85

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 i

JEE Main 2023 Solution View → 86

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is _ _ _ _ _

JEE Main 2023 Solution View → 87

Number of integral solutions to the equation x + y + z = 21 , where x ≥ 1 , y ≥ 3 , z ≥ 4 , is equal to _____ .

JEE Main 2023 Solution View → 88

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

JEE Main 2023 Solution View → 89

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 _____ .

JEE Main 2023 Solution View → 90

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 o

JEE Main 2023 Solution View → 91

The number of ways of selecting two numbers a and b , a ∈ 2 , 4 , 6 , … … , 100 and b ∈ 1 , 3 , 5 , … … , 99 such that 2 is the remainder when a

JEE Main 2023 Solution View → 92

The number of seven digits odd numbers, that can be formed using all the seven digits 1 ,   2 ,   2 ,   2 ,   3 ,   3 ,   5 is

JEE Main 2023 Solution View → 93

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

JEE Main 2023 Solution View → 94

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 :

JEE Main 2023 Solution View → 95

The total number of 4 -digit numbers whose greatest common divisor with 54 is 2 , is

JEE Main 2023 Solution View → 96

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 di

JEE Main 2023 Solution View → 97

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,

JEE Main 2023 Solution View → 98

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1 , 3 , 5 , 7 , 9 without repetition, is

JEE Main 2023 Solution View → 99

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 orange, at least

JEE Main 2023 Solution View → 100

Let f ( x ) = 2 x n + λ , λ ∈ ℝ , n ∈ ℕ , and f ( 4 ) = 133 , f ( 5 ) = 255 . Then the sum of all the positive integer divisors of ( f ( 3 ) - f (

JEE Main 2023 Solution View → 101

Let x and y be distinct integers where 1 ≤ x ≤ 25 and 1 ≤ y ≤ 25 . Then, the number of ways of choosing x and y , such that x + y is divisible by 5 , is ___

JEE Main 2023 Solution View → 102

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 _____ .

JEE Main 2023 Solution View → 103

The number of integers, greater than 7000 that can be formed, using the digits 3 , 5 , 6 , 7 , 8 without repetition is

JEE Main 2023 Solution View → 104

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

JEE Main 2023 Solution View → 105

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 ______

JEE Main 2023 Solution View → 106

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

JEE Advanced 2022 Solution View → 107

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

JEE Advanced 2022 Solution View → 108

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 _____.

JEE Main 2022 Solution View → 109

Let S = 4 , 6 , 9 and T = 9 , 10 , 11 , … , 1000 . If A = a 1 + a 2 + … + a k : k ∈ N , a 1 , a 2 , a 3 , … , a k ∈ S then the sum of all the elements

JEE Main 2022 Solution View → 110

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 + 3 g is equal to

JEE Main 2022 Solution View → 111

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

JEE Main 2022 Solution View → 112

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 num

JEE Main 2022 Solution View → 113

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

JEE Main 2022 Solution View → 114

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 'MANKI

JEE Main 2022 Solution View → 115

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

JEE Main 2022 Solution View → 116

Let b 1 b 2 b 3 b 4 be a 4 -element permutation with b i ∈ 1 , 2 , 3 , … … … , 100 for 1 ≤ i ≤ 4 and b i ≠ b j for i ≠ j , such that

JEE Main 2022 Solution View → 117

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 at

JEE Main 2022 Solution View → 118

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

JEE Main 2022 Solution View → 119

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 __

JEE Main 2022 Solution View → 120

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

JEE Main 2022 Solution View →
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