Combinations Calculator C(n,k)
Calculate combinations C(n,k): how many ways you can choose k items from a set of n, where order does not matter. The basis of combinatorics.
C(n,k)
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Combinations C(n,k): formula and examples
A combination of n objects taken k at a time, with order not mattering, is the binomial coefficient C(n,k) = n! / (k!·(n−k)!). A few familiar examples from Brazilian lotteries and card games make it concrete. Mega-Sena (6 of 60) has C(60,6) = 50,063,860 possible tickets. Lotofácil (15 of 25) comes out to C(25,15) = 3,268,760. A 5-card poker hand drawn from a 52-card deck gives C(52,5) = 2,598,960. A handful of identities come up often: symmetry, C(n,k) = C(n,n−k); the sum over all k from 0 to n, ΣC(n,k) = 2ⁿ, which counts the subsets of an n-set; and Vandermonde's identity, C(m+n,k) = Σ C(m,i)·C(n,k−i).
Applications: probability, lotteries and ENEM
Combinations sit under classical probability (Laplace's favorable-over-possible cases) and under the betting odds in lotteries and poker, and they are standard fare in Brazilian college-entrance exams like the ENEM and vestibular. The same counting shows up when you pick a committee, build a tasting menu, draw a sample in statistics or read off Pascal's triangle.
FAQ
Combination or permutation? When the order of what you picked makes no difference (a lottery draw, a committee, a hand of cards), reach for a combination. When order does change the outcome (a podium, a password), use a permutation.
Why does C(n,k) = C(n,n−k)? Choosing which k go in is the very same choice as deciding which n−k stay out.
What if k = 0 or k = n? Both come out to 1. There is exactly one way to take nothing and one way to take everything.
What if k > n? Then C(n,k) = 0. You can't pick more elements than there are to begin with.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.