68 free MCQs on Permutations and Combinations, each with its own worked answer and explanation. Counting, arrangements, and selections
Take the timed Permutations and Combinations chapterwise test →68 practice questions on Permutations and Combinations, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Permutations and Combinations notes.
Counting tree for selecting 2 items from {A, B, C} without repetition: 3 x 2 = 6 ordered arrangements.
Q3.
nPr represents:
Q4.
nCr represents:
Q12.
How many ways can we choose 2 out of 6 flavors of ice cream?
Q14.
nPr = r! × nCr. True or false?
Q15.
How many 2-digit numbers can be formed from digits 1,2,3,4,5 without repetition?
Q16.
In how many ways can a committee of 3 be chosen from 7?
Q17.
Multiplication principle: if 3 choices for first and 4 for second, total combinations:
Q18.
How many 3-letter words from ABCDE (letters not repeated)?
Q20.
A coin is tossed 3 times. Number of ways to get exactly 2 heads:
Q22.
Number of ways to arrange letters in MISSISSIPPI:
Q23.
Number of ways to select at least one from n objects:
Q24.
Using binomial theorem, coefficient of x³ in (1+x)<sup>7</sup> is:
Q25.
Sum of all coefficients in (x+y)<sup>n</sup> is:
Q27.
How many 4-digit numbers can be formed from 1-9 without repetition?
Q28.
In how many ways can a team of 3 boys and 2 girls be chosen from 5 boys and 4 girls?
Q30.
The number of triangles formed by 10 points (no 3 collinear):
Q31.
How many words can be formed from GARDEN using all letters?
Q32.
ABCD... if A must be first: arrangements of 5 people ABCDE with A first:
Q33.
Number of ways to distribute 5 identical balls into 3 distinct boxes:
Q34.
The number of permutations of n things taken all at once when p are alike:
Q35.
How many ways can 10 people be divided into two groups of 5?
Q36.
Words from BOOK (all permutations):
Q37.
Total number of functions from set A (3 elements) to set B (4 elements):
Q38.
Number of injective (one-to-one) functions from A (3 elements) to B (4 elements):
Q39.
In how many ways can 8 people be arranged in 2 rows of 4?
Q40.
The middle term in the expansion of (1+x)<sup>2n</sup> is:
Q41.
The number of ways to distribute n identical objects into r distinct groups (some may be empty):
Q42.
Number of non-negative integer solutions to x₁ + x₂ + x₃ = 10:
Q43.
Derangement D(n): number of permutations where no element is in original position. D(4) =
Q44.
Total number of permutations of n items with repetitions: n₁ alike, n₂ alike,..., nk alike (sum = n) =
Q45.
Number of ways to seat 5 couples in a row such that each couple sits together:
Q46.
In how many ways can 4 red, 3 blue, and 2 green balls be arranged in a row?
Q47.
Number of ways to select 3 from 6 where order matters for first 2 but not the third:
Q49.
The number of lattice paths from (0,0) to (m,n) moving only right or up:
Q50.
Number of binary strings of length n with exactly k ones:
Q51.
Inclusion-Exclusion: |A ∪ B ∪ C| =
Q53.
Number of ways to arrange n people in a circle with one fixed (labeled) position:
Q54.
In a group of 10 people, handshakes if everyone shakes everyone else exactly once:
Q56.
Total number of onto functions from A (m elements) to B (n elements) using inclusion-exclusion:
Q57.
The Stirling number S(n,k) counts:
Q58.
Number of ways to choose a committee of 5 from 10 where at least 2 women (4 women, 6 men):
Q59.
The generating function for combinations is related to:
Q61.
The number of distinct arrangements of the letters of the word MISSISSIPPI is:
Q62.
The number of 4-digit numbers with all distinct digits chosen from 1–9 that are even is:
Q63.
The number of diagonals of a convex 12-sided polygon is:
Q64.
The number of ways to seat 5 boys and 3 girls in a row so that no two girls are adjacent is:
Q65.
From 6 men and 4 women, the number of 5-member committees with at least 2 women is:
Q66.
The sum of all 4-digit numbers formed using each of 1, 2, 3, 4 exactly once is:
Q67.
The number of ways to distribute 10 identical balls into 3 distinct boxes with each box non-empty is:
Q68.
The number of positive integers less than 1000 divisible by neither 5 nor 7 is:
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