Exercise 6.
Based on the previous question, give a combinatorial proof for the identity:
\begin{equation*}
{n \choose k} = {n+k-1 \choose k} - \sum_{j=1}^n (-1)^{j+1}{n \choose j}{n+k-(2j+1) \choose k - 2j}\text{.}
\end{equation*}
Based on the previous question, give a combinatorial proof for the identity: