Permutation & Combination Calculator

Count permutations (order matters) and combinations (order doesn't), with or without repetition, exactly.

Count	Choose r from n (P and C)
Total items (n)	10
Chosen (r)	3
Repetition allowed	No

Count

Total items (n)

Chosen (r)

Repetition allowed

Combinations (order doesn't matter)

120

Choosing 3 from 10: 120 combinations, or 720 arrangements if order matters.

Exact. Checked exactly.

Method: n!/(n − r)! and n!/(r!(n − r)!)
Worked out with: BigInt
Check: Every step recomputed with exact fractions

Permutations (order matters): 720
Combinations: 120

How this was worked out

1
Permutations
P(n, r) = n! ÷ (n − r)! = 10! ÷ 7!
2
Combinations
C(n, r) = n! ÷ (r!(n − r)!) = 10! ÷ (3! × 7!)

About the Permutation & Combination Calculator

A permutation counts ordered arrangements: the first, second and third places in a race. A combination counts unordered selections: a 3-person committee. For the same n and r there are always r! times as many permutations as combinations. Results are exact even when they run to hundreds of digits.

Restrictions change the count: letters that repeat (n!/(p!q!…), as in MISSISSIPPI), people who must sit together (treat them as one block), people who must not sit together (place the others first, then use the gaps) and circular arrangements ((n − 1)!, halved for necklaces).

The formula

P(n, r) = n! ÷ (n − r)!; C(n, r) = n! ÷ (r!(n − r)!)

With repetition: P = nʳ, C = C(n + r − 1, r)
n!: n × (n − 1) × … × 1

Worked example

Lottery: choose 6 from 49

C(49, 6) = 13,983,816
Chance of one ticket winning = 1 in 13,983,816

Questions people ask

Is a combination lock really a combination?

No: order matters, so it's a permutation (with repetition).

What's C(n, r) called?

“n choose r”, a binomial coefficient.