Coefficient of $$x^r$$ in $$(1+x)^m(1+x)^n$$

Coefficient of $$x^r$$ in $$(1+x)^m(1+x)^n$$

## Formula

Since:

$$(1+x)^m(1+x)^n=(1+x)^{m+n}$$

the coefficient of $x^r$ is:

$${}^{m+n}C_r$$

Therefore:

$$\sum_{k=0}^{r}{}^mC_k{}^nC_{r-k}={}^{m+n}C_r$$

## Usage

- Used to convert complicated coefficient sums into a single binomial coefficient.