Combination Calculator
What is a combination?
A combination C(n, k) is the number of ways to choose k items from n when order does not matter. The same quantity is also written nCr or C(n, r); this page uses k to match the inputs.
| Symbol | Meaning |
|---|---|
| n | Total number of items |
| k | Number of items selected |
| C(n, k) | Number of combinations |
If order matters, that is a permutation: P(n, k) = C(n, k) x k!. For n! see the factorial calculator.
Formula
C(n, k) = n! / (k! x (n-k)!)
C(n, k) = C(n, n-k). This calculator uses k everywhere.
Limitations
k must be an integer between 0 and n; k > n is invalid, not zero. n is capped at 170. Direct n! overflows quickly; the engine multiplies the k terms (n x (n-1) x ... x (n-k+1)) / k! instead of building full factorials.
Sampling with replacement is a different formula and is not modeled here.
FAQ
What is C(n, k)?
Ways to choose k items from n without regard to order. Also written nCr.
Combination vs permutation?
Combinations ignore order. Permutations count each ordering: P(n, k) = C(n, k) x k!.
What is C(52, 5)?
2,598,960 five-card poker hands. Use the poker preset above to load it.