Combinations & Permutations Calculator
How many ways to choose or arrange r items from a set of n.
- Combinations (nCr)120
- Permutations (nPr)720
Inputs used
- Total items (n)10
- Items chosen (r)3
About this calculator
Combinations count selections where order doesn't matter; permutations count arrangements where it does. Choosing 3 pizza toppings from 10 is a combination (₁₀C₃ = 120); awarding gold, silver and bronze among 10 runners is a permutation (₁₀P₃ = 720).
The formulas: nPr = n! ÷ (n−r)!, and nCr = nPr ÷ r! — dividing by r! removes the orderings of the chosen items, which is why the permutation count is always r! times the combination count.
r can't exceed n (you can't choose more items than exist). The numbers explode quickly: a 52-card deck has more orderings (52!) than there are atoms visible in the universe.
Quick reference
Combinations (nCr) by n and r
The number of ways to choose r items (columns) from n items (rows), where order does not matter.
| Total items (n) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 5 | 5 | 10 | 10 | 5 |
| 6 | 6 | 15 | 20 | 15 |
| 7 | 7 | 21 | 35 | 35 |
| 8 | 8 | 28 | 56 | 70 |
| 9 | 9 | 36 | 84 | 126 |
| 10 | 10 | 45 | 120 | 210 |
| 11 | 11 | 55 | 165 | 330 |
| 12 | 12 | 66 | 220 | 495 |
Frequently asked questions
What is the difference between a combination and a permutation?
A combination counts selections where order does not matter; a permutation counts arrangements where it does. Choosing 3 toppings from 10 is a combination (₁₀C₃ = 120), but ranking 3 of 10 runners is a permutation (₁₀P₃ = 720).
What is 5 choose 2?
Ten. Using nCr = n! ÷ (r!(n−r)!), ₅C₂ = 120 ÷ (2 × 6) = 10 — the number of distinct pairs you can pick from five items.
What is the formula for combinations?
nCr = n! ÷ (r! × (n − r)!). Equivalently it is the permutation count nPr divided by r!, which removes the different orderings of the r chosen items.
Why is nPr always at least as large as nCr?
Because permutations count every ordering of the chosen items separately, while combinations count each selection once. Exactly r! orderings collapse into one combination, so nPr = nCr × r! and the two are equal only when r is 0 or 1.
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