binomial coefficient
A binomial coefficient counts the number of ways to choose k items from n distinct items when the order of the chosen items does not matter. It is read “n choose k” and written C(n, k) or as a two-row stacked symbol.
Its formula is C(n, k) = n! / (k!·(n − k)!), where the exclamation marks are factorials. The same numbers also appear as the coefficients in the binomial theorem and as the entries of Pascal's triangle, which is no coincidence — choosing terms when you expand (a + b)^n is exactly a counting problem.
Two handy facts: C(n, 0) = C(n, n) = 1 (there is one way to choose nothing and one way to choose everything), and C(n, k) = C(n, n − k) (choosing what to keep is the same as choosing what to leave out). Because it counts, a binomial coefficient is always a whole number, even though its formula is a fraction.
C(5, 2) = 5! / (2!·3!) = 120 / (2·6) = 10 — there are 10 ways to choose 2 of 5 things.
The fractional formula always returns a whole number.