<s>[INST] Prove $\binom{2n+2}{n+1} = \binom{2n}{n+1} + 2\binom{2n}{n} + \binom{2n}{n-1}$ I need some help showing that these are equivalent. I made a couple attempts to get this right but so far the following work is as far as I've gotten.
Here is the question in its entirety:

Let n be a natural number. Give a combinatorial proof of the following:
  $\binom{2n+2}{n+1} = \binom{2n}{n+1} + 2\binom{2n}{n} + \binom{2n}{n-1}$

My first impression was that I could use the $\binom{n}{k} = \binom{n}{n-k}$ identity to make the term "$\binom{2n}{n-1}$" equal "$\binom{2n}{n+1}$" so I could simplify the equation a bit. I now have:
$\binom{2n+2}{n+1} = 2\binom{2n}{n+1} + 2\binom{2n}{n} = 2[\binom{2n}{n+1} + \binom{2n}{n}]$
Afterwards i figured I could use pascals identity $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ to get rid of yet more terms resulting in:
$\binom{2n +2}{n+1} = 2\binom{2n+1}{n+1}$
This is where I realize I made a pretty big mistake and got stuck, I'm fairly sure I made a maths error somewhere but I am unsure where after my third attempt at this.
 [/INST] \begin{eqnarray*}
 \binom{2n+2}{n+1}= \color{red}{\binom{2n+1}{n+1}}+\color{blue}{\binom{2n+1}{n}}=\color{red}{\binom{2n}{n+1}+\binom{2n}{n}}+\color{blue}{\binom{2n}{n}+\underbrace{\binom{2n}{n-1}}_{\binom{2n}{n-1}=\binom{2n}{n+1}}}=2 \left(\binom{2n}{n+1}+\binom{2n}{n}\right)
\end{eqnarray*}
</s>