JEE Mathematics

Applications of P&C: JEE Questions & Solutions

81 previous-year JEE Mathematics questions on Applications of P&C, each with a worked step-by-step solution.

Advanced

Previous-year questions

  1. 01The 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,…MCQMediumJEE 2026
  2. 02Let S = {(m, n): m, n ∈ {1, 2, 3, …, 50}}. If the number of elements (m, n) in S such that 6^m+9ⁿ is a multiple of 5 is p and the number of…NVAMediumJEE 2026
  3. 03The largest n ∈ ℕ, for which 7ⁿ divides 101!, is:MCQEasyJEE 2026
  4. 04If ( 1/¹⁵C₀ + 1/¹⁵C₁) ( 1/¹⁵C₁ + 1/¹⁵C₂) … ( 1/¹⁵C₁₂ + 1/¹⁵C₁₃) = α¹³/(¹⁴C₀ ¹⁴C₁ … ¹⁴C₁₂), then 30α is equal to.NVAMediumJEE 2026
  5. 05Let ABC be a triangle. Consider four points p₁, p₂, p₃, p₄ on the side AB, five points p₅, p₆, p₇, p₈, p₉ on the side BC, and four points…NVAMediumJEE 2026
  6. 06The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is:NVAMediumJEE 2026
  7. 07The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is:MCQEasyJEE 2026
  8. 08Let 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…NVAMediumJEE 2026
  9. 09The 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 repetition of…NVAMediumJEE 2026
  10. 10The largest value of n, for which 40ⁿ divides 60!, isMCQEasyJEE 2026
  11. 11The letters of the word UDAYPUR'' are written in all possible ways with or without meaning and these words are arranged as in a dictionary.…MCQMediumJEE 2026
  12. 12Let 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…MCQMediumJEE 2026
  13. 13Three persons enter a lift at the ground floor. The lift will go up to the 10th floor. The number of ways in which the three persons can…NVAEasyJEE 2026
  14. 14Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f: A → B such that 1 ∈ f(A) is equal to:MCQMediumJEE 2025
  15. 15In 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…MCQMediumJEE 2025
  16. 16If Σ_(r=1)³⁰ (r² ( C(30,r))²)/C(30,r-1) = α × 2²⁹, then α is equal to.NVAMediumJEE 2025
  17. 17The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, is:MCQEasyJEE 2025
  18. 18The 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:NVAMediumJEE 2025
  19. 19The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is.NVAMediumJEE 2025
  20. 20Number of functions f: {1, 2, …, 100} → {0, 1}, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:NVAEasyJEE 2025
  21. 21The 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…MCQMediumJEE 2025
  22. 22The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is.NVAMediumJEE 2025
  23. 23Let P be the set of seven-digit numbers with the sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2,…MCQMediumJEE 2025
  24. 24The 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…NVAMediumJEE 2025
  25. 25If 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…MCQMediumJEE 2025
  26. 26The largest n ∈ ℕ such that 3ⁿ divides 50! is:MCQEasyJEE 2025
  27. 27The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure described below so that no row remains…MCQMediumJEE 2025
  28. 28If the number of seven-digit numbers, such that the sum of their digits is even, is m · n · 10ⁿ; m, n ∈ {1, 2, 3,..., 9}, then m + n is…NVAEasyJEE 2025
  29. 29Line L₁ of slope 2 and line L₂ of slope 1/2 intersect at the origin O. In the first quadrant, P₁, P₂, …, P₁₂ are 12 points on line L₁ and…MCQEasyJEE 2025
  30. 30Consider the sets A = {(x, y) ∈ ℝ × ℝ: x² + y² = 25}, B = {(x, y) ∈ ℝ × ℝ: x² + 9y² = 144}, C = {(x, y) ∈ ℤ × ℤ: x² + y² ≤ 4}, and D = A ∩…MCQMediumJEE 2025
  31. 31Let m and n, m < n be two 2-digit numbers. Then the total number of pairs (m, n) such that gcd(m, n) = 6, isNVAMediumJEE 2025
  32. 32From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4…MCQMediumJEE 2025
  33. 33For n ≥ 2, let Sₙ denote the set of all subsets of {1, 2, 3, …, n} with no two consecutive numbers. For example, {1, 3, 5} ∈ S₆, but {1, 2,…NVAMediumJEE 2025
  34. 34Let 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…MCQMediumJEE 2025
  35. 35There 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…MCQEasyJEE 2025
  36. 36Let α = (4!)!/(4!)^3! and β = (5!)!/(5!)^4!. Then:MCQEasyJEE 2024
  37. 37All the letters of the word “GTWENTY” are written in all possible ways with or without meaning, and these words are arranged as in a…MCQMediumJEE 2024
  38. 38If ¹¹C₁/2 + ¹¹C₂/3 + … + ¹¹C₉/10 = n/m with gcd(n, m) = 1, then n + m is equal to:MCQMediumJEE 2024
  39. 39The number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to:MCQMediumJEE 2024
  40. 40In an examination of Mathematics paper, there are 20 questions of equal marks, and the question paper is divided into three sections: A, B,…NVAMediumJEE 2024
  41. 41The 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…MCQMediumJEE 2024
  42. 42The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples is:MCQEasyJEE 2024
  43. 43If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of…MCQMediumJEE 2024
  44. 44There are 5 points P₁, P₂, P₃, P₄, P₅ on the side AB, excluding A and B, of a triangle ABC. Similarly, there are 6 points P₆, P₇, …, P₁₁ on…MCQEasyJEE 2024
  45. 45The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is:MCQEasyJEE 2023
  46. 46The 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…MCQEasyJEE 2023
  47. 47Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S such that the sum of their elements is divisible by 3 is.NVAMediumJEE 2023
  48. 48Let 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…NVAMediumJEE 2023
  49. 49Σ_(k=0)⁶ ^(51-k)C₃: is equal toMCQEasyJEE 2023
  50. 50If all the six-digit numbers x₁x₂x₃x₄x₅x₆ with 0 < x₁ < x₂ < x₃ < x₄ < x₅ < x₆ are arranged in increasing order, then the sum of the digits…NVAMediumJEE 2023
  51. 51Five-digit numbers are formed using the digits {1, 2, 3, 5, 7} with repetitions allowed, and are written in descending order with serial…MCQMediumJEE 2023
  52. 52The 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…MCQMediumJEE 2023
  53. 53The number of 3-digit numbers that are divisible by either 3 or 4 but not divisible by 48 is:MCQMediumJEE 2023
  54. 54The number of ways of selecting two numbers a and b, a ∈ {2, 4, 6, …, 100} and b ∈ {1, 3, 5, 7, …, 99} such that 2 is the remainder when a…MCQMediumJEE 2023
  55. 55The number of seven-digit odd numbers that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is........NVAMediumJEE 2023
  56. 56Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to:NVAEasyJEE 2023
  57. 57Let A = [a_ij], where a_ij ∈ ℤ ∩ [0, 4], 1 ≤ i, j ≤ 2. The number of matrices A such that the sum of all entries is a prime number p ∈ (2,…NVAMediumJEE 2023
  58. 58The value of 1/1!50! + 1/3!48! + 1/5!46! + … + 1/49!2! + 1/51!1! is:MCQMediumJEE 2023
  59. 59The number of 3-digit numbers that are divisible by either 2 or 3 but not divisible by 7 is:NVAMediumJEE 2023
  60. 60The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur…NVAMediumJEE 2023
  61. 61The total number of six-digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is.NVAMediumJEE 2023
  62. 62Number of integral solutions to the equation x + y + z = 21, where x ≥ 1, y ≥ 3, z ≥ 4, is.NVAEasyJEE 2023
  63. 63The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is:NVAMediumJEE 2023
  64. 64All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers.…MCQMediumJEE 2023
  65. 65The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters…NVAEasyJEE 2023
  66. 66The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is:MCQMediumJEE 2023
  67. 67The 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:MCQMediumJEE 2023
  68. 68The largest natural number n such that 3ⁿ divides 66! is:NVAEasyJEE 2023
  69. 69Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple is in a match, is…NVAMediumJEE 2023
  70. 70The number of permutations of the digits 1, 2, 3,..., 7 without repetition, which neither contain the string 153 nor the string 2467, is:NVAMediumJEE 2023
  71. 71Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most 3 persons,…MCQMediumJEE 2023
  72. 72The number of triplets (x, y, z), where x, y, z are distinct non-negative integers satisfying x + y + z = 15, is:MCQMediumJEE 2023
  73. 73In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students…MCQEasyJEE 2023
  74. 74If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial number, then the…MCQMediumJEE 2023
  75. 75The 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…MCQMediumJEE 2023
  76. 76Let 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…NVAMediumJEE 2023
  77. 77The number of seven digit positive integers formed using the digits 1, 2, 3, and 4 only and the sum of the digits equal to 12 is:NVAMediumJEE 2023
  78. 78All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written in a dictionary with serial…MCQMediumJEE 2023
  79. 79Total 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:NVAMediumJEE 2023
  80. 80The 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…MCQMediumJEE 2023
  81. 81A 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…NVAMediumJEE 2023

Frequently asked questions

Our bank has 81 previous-year JEE Mathematics questions tagged to Applications of P&C, drawn from JEE papers between 2023 and 2026, including 13 from 2026 alone. Each comes with a full step-by-step solution.

Applications of P&C is rated Advanced overall for JEE Mathematics. The PYQs on this page break down as 60 medium, 21 easy.

Of the 81 previous-year questions on Applications of P&C, 45 are multiple-choice (MCQ) and 36 are numerical-value (NVA) questions.

Based on recent paper analysis, Applications of P&C is worth about 5 marks in JEE Main. It sits in Permutations & Combinations, a Class 11 Mathematics chapter.

More Permutations & Combinations topics

Combinations (C(n,r))8Permutations (P(n,r))3Fundamental Principle of Counting1

Master Applications of P&C with AI-adaptive practice.

Practice unlimited Applications of P&C questions, track your mastery, and get every step explained by an AI tutor.

Start free trial