# The Generalized Binomial Coefficient Formula

Recall from the Binomial Coefficients and The Binomial Theorem pages that we say $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ is a binomial coefficient because for the binomial $(x + y)^n$ we have that $\binom{n}{k}$ is the coefficient attached to the simplified term $x^{n-k}y^k$ after the expansion of this binomial since for all $x, y \in \mathbb{R}$ and $n \in \{0, 1, 2, ... \}$ we saw that:

(1)We will now briefly touch upon the generalized binomial coefficient formula which still allows us to compute $\binom{n}{k}$ for $n, k \in \{0, 1, 2, ..., n \}$ and $0 \leq k \leq n$, however, it will allow $n$ to be any real number and for $k$ to be any integer. For $n \in \mathbb{R}$ and $k \in \mathbb{Z}$, the generalized binomial coefficient formula is:

(2)Let's verify the formula above gives us the same value as $\binom{5}{3} = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!} = 10$. We have that $n = 5$ and $k = 3$ and so with the formula we get:

(3)