Combination Calculator – C(n,r) Selections
Calculate the number of combinations C(n,r): ways to choose r items from n distinct items when order does not matter. Enter n and r to get the result. Useful for probability, lottery odds, and counting problems.
Combination Calculator
Calculate C(n,r) = n! / (r! × (n−r)!) — the number of ways to choose r items from n distinct items when order does not matter.
Combination formula
C(n,r) = n! / (r! × (n−r)!)
n = total items, r = items to choose. Order does not matter. Also written as nCr or (n choose r). Example: C(5,2) = 10 ways to pick 2 from 5.
Understanding combinations and when to use them
A combination counts the number of ways to select a group of items when order does not matter. If you have 5 friends and want to invite 2 to dinner, choosing Alice and Bob is the same as choosing Bob and Alice. The number of such selections is C(5,2) = 5!/(2!×3!) = 120/(2×6) = 10.
The formula C(n,r) = n! / (r!(n-r)!) can be derived from permutations. There are P(n,r) ways to arrange r items from n. Each combination of r items can be arranged in r! ways. So P(n,r) = C(n,r) × r!, hence C(n,r) = P(n,r) / r! = n! / (r!(n-r)!).
Combinations appear in lottery odds (how many ways to pick 6 numbers from 49? C(49,6)), committee selection (how many 3-person committees from 10 people? C(10,3)), and binomial coefficients in algebra. Use our combination calculator for quick results. For arrangements where order matters, use the permutation calculator instead.