Proof binomial coefficient

Question

I'm trying to prove the following: $$\binom{n + p}{k} = \sum_{j=0}^n \binom{n}{j} \cdot \binom{p}{k - j}$$ How do I do it? Induction? And can someone hint me at how to start?

Answer 1

In totaly, you have $n+p$ items, you keep $n$ items in box 1 and keep $p$ items in box 2.

You have to pick a total of $k$ items.

If you pick $j$ items from box 1, you have to pick $k-j$ items from box 2 to make it up.