The binomial coefficient of n and k is defined to be n!/k!(n-k)!. It is often written as

( n )
( k )

Well, not exactly like that: basically it's a over k with parentheses around the pair, and no line between them. It's also written _{n}C_{k} and C(n,k) and with other conventions. It's pronounced "n choose k," because it is the number of combinations of n things taken k at a time. That is, it's the number of different ways of selecting k things out of a set of n of them, where the order you take them doesn't matter. So 52 choose 5 is the number of different possible poker hands there are.

It is also the formula for the coefficients in binomial expansion (hence the name). That is:

(x+y)^{n} = _{n}C_{0}*x^{0}y^{n} +
_{n}C_{1}*x^{1}y^{n-1} +
_{n}C_{2}*x^{2}y^{n-2}+ ... +
_{n}C_{k}*x^{k}y^{n-k}+ ... +
_{n}C_{n}*x^{n}y^{0}

(This is the famed binomial theorem)

There are many identities involving these coefficients, and they pop up everywhere in combinatorics and probability.

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