nCr Combination Calculator C(n,r)

C(n, r) Result

What is a Combination?

A combination counts the number of ways to select r items from n items where order does NOT matter. C(n,r) is also called "n choose r" or the binomial coefficient.

C(n, r) = n! / (r! × (n − r)!)
C(10, 3) = 10! / (3! × 7!) = 120

Combinations vs Permutations

C(n,r) = P(n,r) / r! — combinations are always fewer or equal to permutations since order is ignored.

  • Choosing a 3-person team from 10: C(10,3) = 120
  • Lottery: picking 6 numbers from 49: C(49,6) = 13,983,816

Frequently Asked Questions

What is C(5,2)? ▼
C(5,2) = 5!/(2!×3!) = (5×4)/(2×1) = 10. There are 10 ways to choose 2 items from 5 without regard to order.
What is the difference between C(n,r) and P(n,r)? ▼
P(n,r) counts ordered arrangements; C(n,r) counts unordered selections. C(n,r) = P(n,r)/r!. Always: C(n,r) ≤ P(n,r).
What is C(n,0) or C(n,n)? ▼
Both equal 1. C(n,0) = 1 (there is exactly one way to choose nothing). C(n,n) = 1 (there is exactly one way to choose all items).