For $n, k \in mathbb{N}, n \geq k \geq 0,$ the binomial coefficient is the coefficient of the $x^k$ term in the polynomial expansion of the binomial power $(1+x)^n.$ It is denoted as $\binom{n}{k}$ and can be expressed through @factorial notation as
$$ \binom{n}{k} = \frac{n!}{k!(n - k)!}. $$
For any complex number $n \in \mathbb{C}$ and any integer $k \geq 0,$ the generalized binomial coefficient is defined by
$$ \binom{n}{k} = \frac{n(n-1)(n-2) \cdots (n - k + 1)}{k!}. $$
The expansion of any nonnegative integer power $n$ of the @binomial $x+y$ is a sum of the form
$$ (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}. $$
A special case of the Binomial Theorem is the case where $y = 1:$
$$ (x+1)^n = \sum_{k=0}^n \binom{n}{k} x^k. $$