Give me the binomial formula

The binomial formula, also known as the binomial theorem, provides a formula for expanding powers of a binomial. For any non-negative integer $n$, the formula is:

$$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$

where $\binom{n}{k}$ is the binomial coefficient, calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

and $n!$ (n factorial) is the product of all positive integers up to $n$.