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Permutation and Combination — JEE Main & Advanced IOQM PYQs

112 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.

112 questionsIOQMSolutions on every page
1

Find the number of maps f: 1,2,3 1,2,3,4,5 such that f(i) f(j) whenever i < j

Olympiad workbook Solution View → 2

For an integer n 3 and a permutation = (p₁, p₂, , p_n ) of 1,2, , n , we say p₁ is a landmark point if 2 I n-1 and (p_ I-1 -p_I ) (p_ I+1 -p_I )>0 . For example, for n=7 , the perm

Olympiad workbook Solution View → 3

Ari chooses 7 balls at random from n balls numbered 1 to n . If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair o

Olympiad workbook Solution View → 4

Let N be the number of ways of distributing 52 identical balls into 4 distinguishable boxes such that no box is empty and the difference between the number of balls in any two of t

Olympiad workbook Solution View → 5

A 12 12 board is divided into 144 unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non attacking if they are not in the same

Olympiad workbook Solution View → 6

Three couples sit for a photograph in 2 rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How

Olympiad workbook Solution View → 7

Find the number of positive integers n such that the highest power of 7 dividing n! is 8 .

Olympiad workbook Solution View → 8

Consider the 10 -digit number M=9876543210 . We obtain a new 10-digit number from M according to the following rule: we can choose one or more disjoint pairs of adjacent digits in

Olympiad workbook Solution View → 9

Let A= 1,2,3,4,5,6,7,8 , B= 9,10,11,12 , 13,14,15,16 and C= 17,18,19,20,21,22,23 , 24 . Find the number of triples (x, y, z) such that x A, y B, z C and x+y+z=36 .

Olympiad workbook Solution View → 10

Ria has 4 green marbles and 8 red marbles. She arranges them in a circle randomly. If the probability that no two green marbles are adjacent is p q where the positive integers p, q

Olympiad workbook Solution View → 11

Consider a permutation (a₁, a₂, a₃, a₄, a₅ ) of 1,2,3,4,5 . We say the 5-tuple (a₁, a₂, a₃, a₄, a₅ ) is flawless if for all 1 i < j < k 5 , the sequence (a_j, a_j, a_k ) is not an

Olympiad workbook Solution View → 12

How many two digit numbers have exactly 4 positive factors? (Here 1 and the number n are also considered as factors of n .)

Olympiad workbook Solution View → 13

Consider the set of all 6 -digit numbers consisting of only 3 digits, a, b, c , where a, b, c are distinct. Suppose the sum of the these numbers is 593999406 . What is the largest

Olympiad workbook Solution View → 14

A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mut

Olympiad workbook Solution View → 15

For n N , consider non-negative integer-valued functions f on 1,2, , n satisfying f(i) f(j) for i>j and _ i=1 ^n(i+f(i))=2023 . Choose n such that _ i=1 ^n f(i) is the least. How m

Olympiad workbook Solution View → 16

An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If N is the number of one hour journeys that end at the starting vertex, find the su

Olympiad workbook Solution View → 17

In the figure below, 4 of the 6 disks are to be colored black and 2 are to be colored white. Two colorings that can be obtained from one another by a rotation or a reflection of th

Olympiad workbook Solution View → 18

The number of four-digit odd numbers having digits 1,2,3,4 , each occurring exactly once, is:

Olympiad workbook Solution View → 19

Consider a string of n 1 's. We wish to place some + signs in between so that the sum is 1000 . For instance, if n =190 , one may put + signs so as to get 11 ninety times and 1 ten

Olympiad workbook Solution View → 20

Given a 2 2 tile and seven dominoes ( 2 1 tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a 2 7 rectangle us

Olympiad workbook Solution View → 21

For any finite non empty set X of integers, let (X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered pairs ( A, B ) of fin

Olympiad workbook Solution View → 22

Unconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 with 1 and 2 appearing on opposite faces. Further, each face is colored either re

Olympiad workbook Solution View → 23

Find the least positive integer n such that there are at least 1000 unordered pairs of diagonals in a regular polygon with n vertices that intersect at a right angle in the interio

Olympiad workbook Solution View → 24

In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from

Olympiad workbook Solution View → 25

In the land of Binary, the unit of currency is called Ben and currency notes are available in denominations 1,2 , 2^2, 2^3, . Bens. The rules of the Government of Binary stipulate

Olympiad workbook Solution View → 26

Let P be a convex polygon with 50 vertices. A set F of diagonals of P is said to be minimally friendly if any diagonal d F intersects at most one other diagonal in F at a point int

Olympiad workbook Solution View → 27

Consider the set S= (a, b, c, d, e): 0 < a < b < c < d < e < 100 where a, b, c, d, e are integers. If D is the average value of the fourth element of such a tuple in the set, taken

Olympiad workbook Solution View → 28

What is the number of triples ( a , b , c ) of positive integers such that (i) a < b < c < 10 and (ii) a , b , c , 10 form the sides of a quadrilateral?

Olympiad workbook Solution View → 29

The smallest positive integer that does not divide 1 2 3 4 5 6 7 8 9 is:

Olympiad workbook Solution View → 30

Let d(m) denote the number of positive integer divisors of a positive integer m . If r is the number of integers n 2023 for which _ i=1 ^n d(i) is odd, find the sum of the digits o

Olympiad workbook Solution View → 31

Consider the grid of points X= (m, n) 0 m, n 4 . We say a pair of points (a, b),(c, d) in X is a knightmove pair if (c=a 2 and d=b 1) or (c=a 1 and d=b 2) . The number of knight-mo

Olympiad workbook Solution View → 32

Let n=2 ¹⁹ 3¹² . Let M denote the number of positive divisors of n^2 which are less than n but would not divide n . What is the number formed by taking the last two digits of M (in

Olympiad workbook Solution View → 33

There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room).

Olympiad workbook Solution View → 34

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a+c=b+d . Let S be any set of quadruples ( a, b, c, d ) where 1 a < b < d < c 20 and where the cardinality o

Olympiad workbook Solution View → 35

The six sides of a convex hexagon A₁ A₂ A₃ A₄ A₅ A₆ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If N is the number of colorings such that e

Olympiad workbook Solution View → 36

In how many ways can four married couples sit in a merry go round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

Olympiad workbook Solution View → 37

In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected

Olympiad workbook Solution View → 38

Find the number of ordered triples (a, b, c) of positive integers such that 30 a+50 b+70 c 343 .

Olympiad workbook Solution View → 39

Initially, there are 3⁸⁰ particles at the origin (0,0) . At each step the particles are moved to points above the x -axis as follows: if there are n particles at any point ( x, y )

Olympiad workbook Solution View → 40

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, s

Olympiad workbook Solution View → 41

We will say that a rearrangement of the letters of a word has no fixed letters if. When the rearrangement is placed directly below the word, no column has the same letter repeated.

Olympiad workbook Solution View → 42

A 1 n rectangle (n 1) is divided into n unit (1 1) squares. Each square of this rectangle is coloured red, blue or green. Let f(n) be the number of colourings of the rectangle in w

Olympiad workbook Solution View → 43

Consider the set E= 5,6,7,8,9 . For any partition A, B of E , with both A and B non empty. Consider the number obtained by adding the product of elements of A to the product of ele

Olympiad workbook Solution View → 44

Suppose in the plane 10 pair wise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

Olympiad workbook Solution View → 45

Find the number of ordered triples (a, b, c) of positive integers such that a b c=108 .

Olympiad workbook Solution View → 46

A binary sequence is a sequence in which each term is equal to 0 or 1 . A binary sequence is called friendly if each term is adjacent to at least one term that is equal to 1 . For

Olympiad workbook Solution View → 47

Determine the number of 8-tuples ( ₁, ₂, , ₈ ) such that ₁, ₂, , ₈ 1,-1 and ₁+2 ₂+3 ₃+ +8 ₈ is a multiple of 3 .

Olympiad workbook Solution View → 48

A positive integer k is said to be good if there exists a partition of 1,2,3, , 20 in to disjoint proper subsets such that the sum of the numbers in each subset of the partition is

Olympiad workbook Solution View → 49

Let N be the number of ways of choosing a subset of 5 distinct numbers from the set 10 a+b: 1 a 5,1 b 5 where a , b are integers, such that no two of the selected numbers have the

Olympiad workbook Solution View → 50

Consider the set F of all polynomials whose coefficients are in the set of 0,1 . Let q(x)=x^3+x+1 . The number of polynomials p(x) in F of degree 14 such that the product p(x) q(x)

Olympiad workbook Solution View → 51

Let N be the number of ways of distributing 8 chocolates of different brands among 3 children such that each child gets at least one chocolate, and no two children get the same num

Olympiad workbook Solution View → 52

What is the number of ways in which one can choose 60 units square from a 11 11 chessboard such that no two chosen square have a side in common ?

Olympiad workbook Solution View → 53

There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after

Olympiad workbook Solution View → 54

If N is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is N/100 ?

Olympiad workbook Solution View → 55

There are several tea cups in the kitchen, some with handle and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exa

Olympiad workbook Solution View → 56

How many 4-digit numbers abcd are there such that a < b < c < d and b - a < c - b < d - c ?

Olympiad workbook Solution View → 57

Let A= 1,2,3,4,5,6,7,8 , B= 9,10,11,12 , 13,14,15,16 and C= 17,18,19,20,21,22,23 , 24 . Find the number of triples (x, y, z) such that x A, y B, z C and x+y+z=36 .

Olympiad workbook Solution View → 58

Find the number of positive integers n such that the highest power of 7 dividing n! is 8 .

Olympiad workbook Solution View → 59

Ria has 4 green marbles and 8 red marbles. She arranges them in a circle randomly. If the probability that no two green marbles are adjacent is p q where the positive integers p, q

Olympiad workbook Solution View → 60

How many two digit numbers have exactly 4 positive factors? (Here 1 and the number n are also considered as factors of n .)

Olympiad workbook Solution View → 61

Three couples sit for a photograph in 2 rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How

Olympiad workbook Solution View → 62

Consider a permutation (a₁, a₂, a₃, a₄, a₅ ) of 1,2,3,4,5 . We say the 5-tuple (a₁, a₂, a₃, a₄, a₅ ) is flawless if for all 1 i < j < k 5 , the sequence (a_j, a_j, a_k ) is not an

Olympiad workbook Solution View → 63

Ari chooses 7 balls at random from n balls numbered 1 to n . If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair o

Olympiad workbook Solution View → 64

Find the number of maps f: 1,2,3 1,2,3,4,5 such that f(i) f(j) whenever i < j

Olympiad workbook Solution View → 65

Let N be the number of ways of distributing 52 identical balls into 4 distinguishable boxes such that no box is empty and the difference between the number of balls in any two of t

Olympiad workbook Solution View → 66

Consider the 10 -digit number M=9876543210 . We obtain a new 10-digit number from M according to the following rule: we can choose one or more disjoint pairs of adjacent digits in

Olympiad workbook Solution View → 67

For an integer n 3 and a permutation = (p₁, p₂, , p_n ) of 1,2, , n , we say p₁ is a landmark point if 2 I n-1 and (p_ I-1 -p_I ) (p_ I+1 -p_I )>0 . For example, for n=7 , the perm

Olympiad workbook Solution View → 68

A 12 12 board is divided into 144 unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non attacking if they are not in the same

Olympiad workbook Solution View → 69

In the figure below, 4 of the 6 disks are to be colored black and 2 are to be colored white. Two colorings that can be obtained from one another by a rotation or a reflection of th

Olympiad workbook Solution View → 70

An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If N is the number of one hour journeys that end at the starting vertex, find the su

Olympiad workbook Solution View → 71

Consider the set of all 6 -digit numbers consisting of only 3 digits, a, b, c , where a, b, c are distinct. Suppose the sum of the these numbers is 593999406 . What is the largest

Olympiad workbook Solution View → 72

Consider a string of n 1 's. We wish to place some + signs in between so that the sum is 1000 . For instance, if n =190 , one may put + signs so as to get 11 ninety times and 1 ten

Olympiad workbook Solution View → 73

The number of four-digit odd numbers having digits 1,2,3,4 , each occurring exactly once, is:

Olympiad workbook Solution View → 74

A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mut

Olympiad workbook Solution View → 75

The smallest positive integer that does not divide 1 2 3 4 5 6 7 8 9 is:

Olympiad workbook Solution View → 76

Consider the set S= (a, b, c, d, e): 0 < a < b < c < d < e < 100 where a, b, c, d, e are integers. If D is the average value of the fourth element of such a tuple in the set, taken

Olympiad workbook Solution View → 77

Consider the grid of points X= (m, n) 0 m, n 4 . We say a pair of points (a, b),(c, d) in X is a knightmove pair if (c=a 2 and d=b 1) or (c=a 1 and d=b 2) . The number of knight-mo

Olympiad workbook Solution View → 78

For n N , consider non-negative integer-valued functions f on 1,2, , n satisfying f(i) f(j) for i>j and _ i=1 ^n(i+f(i))=2023 . Choose n such that _ i=1 ^n f(i) is the least. How m

Olympiad workbook Solution View → 79

Let d(m) denote the number of positive integer divisors of a positive integer m . If r is the number of integers n 2023 for which _ i=1 ^n d(i) is odd, find the sum of the digits o

Olympiad workbook Solution View → 80

Given a 2 2 tile and seven dominoes ( 2 1 tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a 2 7 rectangle us

Olympiad workbook Solution View → 81

The six sides of a convex hexagon A₁ A₂ A₃ A₄ A₅ A₆ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If N is the number of colorings such that e

Olympiad workbook Solution View → 82

For any finite non empty set X of integers, let (X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered pairs ( A, B ) of fin

Olympiad workbook Solution View → 83

Find the least positive integer n such that there are at least 1000 unordered pairs of diagonals in a regular polygon with n vertices that intersect at a right angle in the interio

Olympiad workbook Solution View → 84

In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from

Olympiad workbook Solution View → 85

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a+c=b+d . Let S be any set of quadruples ( a, b, c, d ) where 1 a < b < d < c 20 and where the cardinality o

Olympiad workbook Solution View → 86

In the land of Binary, the unit of currency is called Ben and currency notes are available in denominations 1,2 , 2^2, 2^3, . Bens. The rules of the Government of Binary stipulate

Olympiad workbook Solution View → 87

Let P be a convex polygon with 50 vertices. A set F of diagonals of P is said to be minimally friendly if any diagonal d F intersects at most one other diagonal in F at a point int

Olympiad workbook Solution View → 88

There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after

Olympiad workbook Solution View → 89

Unconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 with 1 and 2 appearing on opposite faces. Further, each face is colored either re

Olympiad workbook Solution View → 90

Initially, there are 3⁸⁰ particles at the origin (0,0) . At each step the particles are moved to points above the x -axis as follows: if there are n particles at any point ( x, y )

Olympiad workbook Solution View → 91

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, s

Olympiad workbook Solution View → 92

What is the number of triples ( a , b , c ) of positive integers such that (i) a < b < c < 10 and (ii) a , b , c , 10 form the sides of a quadrilateral?

Olympiad workbook Solution View → 93

Let n=2 ¹⁹ 3¹² . Let M denote the number of positive divisors of n^2 which are less than n but would not divide n . What is the number formed by taking the last two digits of M (in

Olympiad workbook Solution View → 94

In how many ways can four married couples sit in a merry go round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

Olympiad workbook Solution View → 95

A binary sequence is a sequence in which each term is equal to 0 or 1 . A binary sequence is called friendly if each term is adjacent to at least one term that is equal to 1 . For

Olympiad workbook Solution View → 96

Suppose in the plane 10 pair wise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

Olympiad workbook Solution View → 97

Find the number of ordered triples (a, b, c) of positive integers such that a b c=108 .

Olympiad workbook Solution View → 98

Find the number of ordered triples (a, b, c) of positive integers such that 30 a+50 b+70 c 343 .

Olympiad workbook Solution View → 99

In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected

Olympiad workbook Solution View → 100

There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room).

Olympiad workbook Solution View → 101

We will say that a rearrangement of the letters of a word has no fixed letters if. When the rearrangement is placed directly below the word, no column has the same letter repeated.

Olympiad workbook Solution View → 102

Consider the set E= 5,6,7,8,9 . For any partition A, B of E , with both A and B non empty. Consider the number obtained by adding the product of elements of A to the product of ele

Olympiad workbook Solution View → 103

A 1 n rectangle (n 1) is divided into n unit (1 1) squares. Each square of this rectangle is coloured red, blue or green. Let f(n) be the number of colourings of the rectangle in w

Olympiad workbook Solution View → 104

Consider the set F of all polynomials whose coefficients are in the set of 0,1 . Let q(x)=x^3+x+1 . The number of polynomials p(x) in F of degree 14 such that the product p(x) q(x)

Olympiad workbook Solution View → 105

Let N be the number of ways of choosing a subset of 5 distinct numbers from the set 10 a+b: 1 a 5,1 b 5 where a , b are integers, such that no two of the selected numbers have the

Olympiad workbook Solution View → 106

How many 4-digit numbers abcd are there such that a < b < c < d and b - a < c - b < d - c ?

Olympiad workbook Solution View → 107

There are several tea cups in the kitchen, some with handle and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exa

Olympiad workbook Solution View → 108

Determine the number of 8-tuples ( ₁, ₂, , ₈ ) such that ₁, ₂, , ₈ 1,-1 and ₁+2 ₂+3 ₃+ +8 ₈ is a multiple of 3 .

Olympiad workbook Solution View → 109

Let N be the number of ways of distributing 8 chocolates of different brands among 3 children such that each child gets at least one chocolate, and no two children get the same num

Olympiad workbook Solution View → 110

What is the number of ways in which one can choose 60 units square from a 11 11 chessboard such that no two chosen square have a side in common ?

Olympiad workbook Solution View → 111

If N is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is N/100 ?

Olympiad workbook Solution View → 112

A positive integer k is said to be good if there exists a partition of 1,2,3, , 20 in to disjoint proper subsets such that the sum of the numbers in each subset of the partition is

Olympiad workbook Solution View →
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