Combination Calculator
Count unordered selections: how many ways to choose r items from n.
What the Combination Calculator does
A combination counts selections where order does not matter — a lottery ticket is the same whichever sequence the numbers are drawn in. It is the permutation count divided by r!, removing the duplicate orderings.
Formula
C(n, r) = n! ÷ (r! × (n − r)!)C(n, r) = C(n, n − r)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Total items (n) | number | Yes | Accepts 0 or more. |
| Items chosen (r) | number | Yes | Accepts 0 or more. |
How to use it
- Enter Total items (n) and Items chosen (r).
- Select Calculate.
Worked example
A 6-from-49 lottery.
- n
- 49
- r
- 6
C(49,6) = 13,983,816, so a single ticket has about a 1 in 14 million chance.
Reading the result
- Combinations count selections where order does not matter, which is why a lottery ticket is the same however the numbers are drawn.
- C(n, r) equals C(n, n − r): choosing which items to take is the same problem as choosing which to leave.
Common mistakes
- Using combinations when the order of selection matters, which undercounts — that case needs permutations.
- Reading C(n, r) as a probability. It is a count of possibilities; the probability is one divided by that count only if every selection is equally likely.
Frequently asked questions
Why does C(n, r) equal C(n, n − r)?
Choosing which 6 to include is the same act as choosing which 43 to leave out.