This page gives three definitions of binomial coefficients in
order of increasing generality. Definition 1: most basic The
most common definition of binomial coefficients is not the most
useful or the most general. Most sources define the binomial
coefficient (n, k) as for where n is a positive integer and 0 ≤
k ≤ n. Call this Definition 1. It turns out to be useful to
replace the above definition with one that allows n to be any
real number and allows k to be any integer. Definition 2: more
general The book Concrete Mathematics defines binomial
coefficients as follows. First, define the kth falling power of
r as The kth falling power of r multiplies starts with r and
multiplies together k numbers, each being one less than its
predecessor. Given this notation, define the binomial
coefficients as Call this Definition 2. Note the use of "r"
rather than "n" to remind us that the top number can be any real
number, not necessarily an integer. When Definition 1 and
Definition 2 both hold, they give the same result.