Permutations and Combinations: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

Start with n! as if all n objects were distinct. But swapping identical items within a group produces no new arrangement. For each group of identical items, divide by the factorial of its group size. So the number of distinct permutations = n! / (p1! x p2! x …).

nCr counts the number of unordered selections of r items chosen from n. Start from nPr, which counts the same items but in every order. Each unordered group of r items can itself be arranged in r! different orders. So divide nPr by r! to remove the order: nCr = n! / (r!(n-r)!).

nPr counts the number of ordered arrangements of r items chosen from n. We start with n! and divide by (n-r)! to remove the leftover items we didn’t arrange. So nPr = n! / (n-r)!.

A permutation is an arrangement where order matters. For example, AB and BA count as two different permutations. A combination is a selection where order does NOT matter. For example, AB and BA count as the same combination. So for the same items, permutations never give fewer outcomes than combinations.

0! means zero factorial, the factorial of zero. Factorials are usually a product of counting numbers, but there is nothing to multiply for 0. By convention, mathematicians define 0! = 1. This convention keeps formulas like nCr and nPr working correctly at their edges.

There are 4 choices for routes to a city and 5 choices for routes back. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 4 x 5 = 20.

There are 5 choices for shirts and 5 choices for trousers. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 5 x 5 = 25.

There are 7 choices for shirts and 9 choices for ties. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 7 x 9 = 63.

There are 9 choices for shirts and 5 choices for ties. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 9 x 5 = 45.

There are 5 choices for shirts and 6 choices for trousers. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 5 x 6 = 30.

There are 4 choices for starters and 5 choices for main courses. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 4 x 5 = 20.

There are 7 choices for starters and 10 choices for main courses. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 7 x 10 = 70.

There are 9 choices for pens and 8 choices for notebooks. These two choices are independent: picking one does not affect the other. By the fundamental counting principle, multiply the number of choices together. Total ways = 9 x 8 = 72.

Choosing the men and choosing the women are two separate, independent steps. 4 men can be chosen from 7 men in 7C4 = 35 ways. 3 women can be chosen from 7 women in 7C3 = 35 ways. By the counting principle, multiply the two counts together for the total. Total ways = 35 x 35 = 1225.

Choosing the men and choosing the women are two separate, independent steps. 2 men can be chosen from 10 men in 10C2 = 45 ways. 4 women can be chosen from 8 women in 8C4 = 70 ways. By the counting principle, multiply the two counts together for the total. Total ways = 45 x 70 = 3150.

Choosing the men and choosing the women are two separate, independent steps. 2 men can be chosen from 11 men in 11C2 = 55 ways. 4 women can be chosen from 8 women in 8C4 = 70 ways. By the counting principle, multiply the two counts together for the total. Total ways = 55 x 70 = 3850.

Choosing the men and choosing the women are two separate, independent steps. 3 men can be chosen from 10 men in 10C3 = 120 ways. 4 women can be chosen from 8 women in 8C4 = 70 ways. By the counting principle, multiply the two counts together for the total. Total ways = 120 x 70 = 8400.

Choosing the men and choosing the women are two separate, independent steps. 3 men can be chosen from 4 men in 4C3 = 4 ways. 2 women can be chosen from 8 women in 8C2 = 28 ways. By the counting principle, multiply the two counts together for the total. Total ways = 4 x 28 = 112.

Choosing the men and choosing the women are two separate, independent steps. 4 men can be chosen from 9 men in 9C4 = 126 ways. 4 women can be chosen from 6 women in 6C4 = 15 ways. By the counting principle, multiply the two counts together for the total. Total ways = 126 x 15 = 1890.

Choosing the men and choosing the women are two separate, independent steps. 3 men can be chosen from 11 men in 11C3 = 165 ways. 1 women can be chosen from 4 women in 4C1 = 4 ways. By the counting principle, multiply the two counts together for the total. Total ways = 165 x 4 = 660.

Choosing the men and choosing the women are two separate, independent steps. 2 men can be chosen from 6 men in 6C2 = 15 ways. 4 women can be chosen from 5 women in 5C4 = 5 ways. By the counting principle, multiply the two counts together for the total. Total ways = 15 x 5 = 75.

Choosing the men and choosing the women are two separate, independent steps. 2 men can be chosen from 4 men in 4C2 = 6 ways. 1 women can be chosen from 10 women in 10C1 = 10 ways. By the counting principle, multiply the two counts together for the total. Total ways = 6 x 10 = 60.

Choosing the men and choosing the women are two separate, independent steps. 4 men can be chosen from 11 men in 11C4 = 330 ways. 3 women can be chosen from 5 women in 5C3 = 10 ways. By the counting principle, multiply the two counts together for the total. Total ways = 330 x 10 = 3300.

One particular person must always be included in the committee. That person’s seat is already decided, so we don’t choose for them. We only need to choose the remaining 3 members from the other 15 people. Number of ways = 15C3 = 455.

One particular person must always be included in the committee. That person’s seat is already decided, so we don’t choose for them. We only need to choose the remaining 3 members from the other 9 people. Number of ways = 9C3 = 84.