Permutation and Combination Calculator

Permutations and Combinations

Permutations and combinations are part of a branch of mathematics called combinatorics, which studies finite, discrete structures.

Permutations are specific selections of elements in which the order matters, while combinations are selections where order does not matter.

For example, a typical “combination” lock should technically be called a permutation lock because the order is important:

• 1-2-9 is different from 2-9-1 (Permutation)
• In a true combination, any order would be the same

The calculator above only considers cases without replacement (no repetition).

Permutations

We calculate r-permutations of n (also called partial permutations) when arranging r items from a set of n elements.

Notation: nPr, P(n,r)

Example: Choosing a captain and goalkeeper from a soccer team of 11 players.

A B C D E F G H I J K  → 11 members
Pick A as captain → 10 left
Pick B as goalkeeper → 9 left (but we stop at 2 picks)

The first choice: 11 options
The second choice: 10 options
Total: 11 × 10 = 110

The general permutation formula:

nPr = ^n!⁄_{(n − r)!}

For this example:

11P2 = ^11!⁄_{(11 − 2)!} = ^11!⁄_9! = 11 × 10 = 110

For permutations with replacement:

nPr = n^r

Combinations

Combinations are similar to permutations, but order does not matter.
Notation: nCr, C(n,r), or binomial coefficient ⁿC_r.

Example: Choosing 2 strikers from a team of 11 players.

Order does not matter — choosing A then B is the same as B then A.

We start with permutations and then remove redundancies by dividing by r!.

The general combination formula:

nCr = ^n!⁄_{r! × (n − r)!}

For this example:

11C2 = ^11!⁄_{2! × (11 − 2)!}
= ^11!⁄_{2! × 9!}
= (11 × 10) ⁄ 2 = 55

For combinations with replacement:

nCr = ^{(r + n − 1)!}⁄_{r! × (n − 1)!}

Summary

• Permutation: Order matters → Formula: ^n!⁄_{(n − r)!}
• Combination: Order does not matter → Formula: ^n!⁄_{r! × (n − r)!}

Related Calculators:
Mean Median Mode Range Calculator, Free Statistics Calculator

External Resources:
Permutation and Combination Calculator on Calculator.net