Compute C(n, k) exactly — 'n choose k' — plus the binomial probability of exactly k successes, with the full distribution table.
Used only for the probability figures. For pure combinations (like poker hands), ignore it and read C(n, k).
binomial coefficient C(n, k)
How to use this tool
n is the total number of items or trials; k is how many you choose, or how many successes you want. C(n, k) is read as 'n choose k'.
For n up to 1000 the coefficient is exact to the last digit (computed with big integers, not floating-point rounding). For larger n it is shown as a power of 10 with its digit count.
If each trial succeeds with probability p, the calculator also gives P(exactly k successes) and P(at most k successes) from the binomial distribution.
Choosing k items is the same as leaving n − k behind, so C(n, k) always equals C(n, n − k). The calculator shows both forms.
Formula & worked example
Worked example: poker hands. C(52, 5) = 52! ÷ (5! × 47!) = 2,598,960 possible five-card hands. And for coin flips: the chance of exactly 3 heads in 10 fair flips is C(10, 3) × 0.5^10 = 120 ÷ 1024 = 0.1172, about 11.72%.
Limitations — read this before you rely on it
Exact coefficients are given for n up to 1000; beyond that the value is shown as a power of 10 because the integer has thousands of digits. Probabilities assume independent trials with a constant p — if trials influence each other, or p changes, the binomial model does not apply.
Frequently asked questions
C(n, k) — 'n choose k' — counts how many ways you can choose k items from n items when order does not matter. Choosing a committee, a lottery line, or a poker hand are all binomial coefficients.
2,598,960 — the number of possible five-card poker hands from a 52-card deck. It is why a royal flush is so rare: only 4 of those hands are royal flushes.
Every choice of k items to take is also a choice of n − k items to leave behind, so the counts must match. The calculator uses the smaller of k and n − k to compute faster.
A permutation counts order: dealing cards one by one. A combination (binomial coefficient) ignores order: the same five cards in any order are one hand. Permutations are larger by a factor of k!.
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k): the number of ways to get k successes, times the probability of one such pattern. Summing k = 0 to your k gives P(at most k).