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 nCk 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 = nC0*x0yn +
nC1*x1yn-1 +
nC2*x2yn-2+ ... +
nCk*xkyn-k+ ... +
nCn*xny0
(This is the famed binomial theorem)
There are many identities involving these coefficients, and they pop up everywhere in combinatorics and probability.
This was a nodeshell