Combinations & Permutations Calculator
Compute nCr and nPr for choosing r from n — enter how many items you have and how many you pick to see both counts, with every step.
Reviewed by the OmniCalc teamMethod verified 2026-07-01
A whole number, 0 or more
A whole number, no more than n
10
Combinations 10, permutations 20- Permutations (nPr)
- 20
- Combinations (nCr)
- 10
Show steps
- Permutations nPr = n! ÷ (n − r)! = 5! ÷ (5 − 2)! = 20.
- Combinations nCr = nPr ÷ r! = 20 ÷ 2! = 10.
How to use the combinations & permutations calculator
- 1Enter the total items (n) — the full set you are choosing from.
- 2Enter how many to choose (r); r must not exceed n.
- 3Read the combinations (nCr) and permutations (nPr), and open Show steps to see the formulas with your numbers.
Which one do I need?
If order matters — say gold, silver and bronze on a podium, or the digits of a PIN — you want permutations (nPr). If only which items you picked matters and their order does not — a lottery ticket or a set of pizza toppings — you want combinations (nCr). The two are linked by nPr = nCr × r!.
Frequently asked questions
What is the difference between a combination and a permutation?
Order. A permutation counts arrangements where the sequence matters, so ABC and CBA are different. A combination counts selections where order is irrelevant, so ABC and CBA are the same group. That is why nPr is always at least as large as nCr — every combination can be arranged in r! different orders.
What are the formulas for nCr and nPr?
Permutations use nPr = n! ÷ (n − r)!, and combinations use nCr = n! ÷ (r! · (n − r)!). Here n is how many items you have, r is how many you pick, and ! is the factorial. The two are linked by nCr = nPr ÷ r!.
What do 'n choose r' and the C(n, r) notation mean?
They are all names for the same number of combinations: 'n choose r', C(n, r), nCr and the binomial coefficient (n over r) each mean the count of unordered ways to pick r items from n. For example, 5 choose 2 is 10.
Why must r be less than or equal to n?
You cannot choose more items than you have. If r is greater than n there are zero ways to make the selection, so the calculator treats r > n (and any negative or fractional count) as undefined rather than returning a misleading number.