Sums of Binomials

Question

I want to understand how to sum binomials and I can not figure out how this comes together:

$$\sum _{l=0}^k \binom{x}{l} \binom{y}{k-l}=\binom{x+y}{k}$$

Answer 1

This will be a duplicate, but the combinatorial argument is:

• Choose a team of $k$ from $x$ girls and $y$ boys by having $l$ girls and $k-l$ boys and summing over possible $l$
• Choose a team of $k$ from from $x$ girls and $y$ boys by treating them as $x+y$ children