So I've been having trouble making combinatorial proofs, as they're hard to start, I just can't simply think easily in the "committee"-way. So this is one of the many properties I have to prove.
$$\binom{n+k}{n+p}\binom{n+p}{p} = \binom{n+k}{n}\binom{k}{p}$$
I just don't know how to start. I'd appreciate if you'd help me and give some advice on how to do combinatorial proofs in general.
In a combinatorial proof, we count the same set of objects in two different ways. Since the two expressions count the same thing, they must be equal.
Let's consider the left-hand side. The factor $\binom{n + k}{n + p}$ counts the number of ways we can select $n + p$ of the $n + k$ available people. The factor $\binom{n + p}{p}$ counts the number of ways we can select $p$ of those $n + p$ people. This suggests we are choosing a committee of $n + p$ people, from which a subcommittee of $p$ people is drawn.
For the right-hand side, we choose $n$ of the $n + k$ people to serve on the committee. From the remaining $k$ people, we select an additional $p$ people to serve on the committee, each of whom will also serve on the subcommittee.