Permutation & Combination Calculator

Find the number of ways to choose r items from n, as permutations (nPr, order matters) and combinations (nCr, order does not matter).

n (total number of items; 0 to 1,000)

r (number chosen)

Permutations nPr (order matters) — Combinations nCr (order does not matter) —

For reference
n! (n factorial)	—

The value copied is the whole number with every digit (no commas). When a result has many digits, an approximation and the number of digits are shown, and you can open "Show all digits" to see every digit.

Permutations and combinations

Permutations (nPr) count the ways to choose r items from n and arrange them. A different order counts as a different arrangement. nPr = n × (n−1) × … × (n−r+1) = n! ÷ (n−r)!
Combinations (nCr) count the ways to choose r items from n. The order does not matter. nCr = nPr ÷ r! = n! ÷ (r! × (n−r)!)
Example: choosing 3 of 10 people for 1st, 2nd and 3rd place gives 720 permutations. Choosing 3 of 10 people for a committee gives 120 combinations.
Relationship: nPr = nCr × r! (arranging the r chosen items in every order turns a combination into a permutation).

Examples

10P3 = 720, 10C3 = 120
52C5 = 2,598,960 (5-card hands from a 52-card deck)
49C6 = 13,983,816 (choosing 6 numbers out of 49)
20P2 = 380, 20C2 = 190
5P5 = 120, 5C5 = 1
1000C500 is a number with 300 digits

When n or r is 0, or r is greater than n

n = 0 or r = 0: there is exactly one way to choose nothing, so the result is 1 (0P0 = 0C0 = 1, nP0 = nC0 = 1, 0! = 1).
n = r: nPn = n!, and nCn = 1 (the one way to choose everything).
r greater than n: you cannot choose more items than you have, so the result is 0 ways (it is not an error).
Negative numbers, decimals and non-numbers are errors.

How it is calculated, and the limits

Results are exact whole numbers, so there are no rounding errors.
n can be up to 1,000 (r is a whole number of 0 or more). 1,000! has 2,568 digits. When a result has 16 or more digits, the calculator shows an approximation and the number of digits, such as "about 4.023873 × 10^2567 (2,568 digits)", and keeps the full number in a collapsed section.
Permutations with repetition (n to the power r) and combinations with repetition are not covered.

Notes

To write a probability as a fraction, use the Fraction Calculator (for example, the number of winning combinations ÷ the number of all combinations).

Frequently asked questions

Should I use nPr or nCr?

Use nPr when order matters (rankings, arrangements, PIN codes) and nCr when order does not matter (choosing a committee, the set of winners in a draw).

Why is 0! equal to 1?

Defining it as 1 (there is one way to arrange nothing) keeps the formulas for nPr and nCr working when n = r or r = 0.

Why is nC0 equal to 1?

There is exactly one way to choose 0 items: choose nothing.