Combination Calculator
Compute the number of ways to choose r items from n distinct items when the
order of selection does not matter — written nCr and read as
"n choose r". Ideal for lottery, committee selection, card hand and team
composition problems.
Enter n and r
Enter n and r, then press Calculate nCr.
The combination formula
nCr = n! / (r! × (n − r)!)
Also known as the binomial coefficient and written
C(n, r), ⁿCᵣ or (n choose r).
Because nCr = nPr / r!, combinations are just permutations
divided by the number of orderings each selection admits. Two selections
that differ only in the order of items count as the same combination.
Key properties
- Symmetry:
nCr = nC(n − r). Choosing 6 of 49 is the same as leaving 43 out of 49. - Boundary:
nC0 = nCn = 1. - Pascal's rule:
nCr = (n − 1)C(r − 1) + (n − 1)Cr.
Worked examples
Example 1 — Committee selection
Choose 3 people from a group of 10 to form a committee (order irrelevant).
10C3 = 10! / (3! × 7!) = 120.
120 committees
Example 2 — Lottery
Choose 6 numbers from 49 (standard 6/49 lottery).
49C6 = 13,983,816 possible tickets.
13,983,816 combinations
Example 3 — Card hand
How many 5-card poker hands can be dealt from a 52-card deck?
52C5 = 2,598,960.
2,598,960 hands
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Frequently asked questions
Use nCr when the order of chosen items does not matter — committee members, lottery tickets, subsets. Use nPr when the order does matter — podium finishes, PIN codes, seating arrangements.
Choosing r items from n is the same as choosing which n − r items to leave out. So nCr = nC(n − r). For example, 10C3 = 10C7 = 120.
nC0 = 1 — there is exactly one way to choose nothing (an empty selection). Similarly, nCn = 1 — one way to choose everything.