Combinatorial proof for the following fact: $\binom{2n}{n+1}+2\binom{2n}{n}+\binom{2n}{n-1}=\binom{2n+2}{n+1}$

Question

$$\binom{2n}{n+1}+2\binom{2n}{n}+\binom{2n}{n-1}=\binom{2n+2}{n+1}$$ Hint: From 2n + 2 players, consider 2 players (call them A, B) separately.

I found this question in my textbook with no context about the players.

Answer 1

Since you've clarified that your question is about the $2\binom{2n}n$ term, I'll relate specifically to that. My advice: don't worry too much about it. If it helps consider it as two separate $\binom{2n}n$ terms added together. These terms should arise organically from your combinatorial interpretation of the problem.

Answer 2

${\mathrm C}^{2n+2}_{n+1}$ counts how to select $n+1$ players from $2n+2$ players (where the hint suggests naming the plus two players A and B).

What then do the three terms $1{\mathrm C}^{2n}_{n+1},2{\mathrm C}^{2n}_{n},$ and $1{\mathrm C}^{2n}_{n-1}$ each count?   How do $1,2,1$ relate to selection of players?   What might the hint about the two named players have to do with it?