Combinações C(n,k)

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.