Exercise 6.
Consider the identity:
\begin{equation*}
k{n\choose k} = n{n-1 \choose k-1}\text{.}
\end{equation*}
Is this true? Try it for a few values of \(n\) and \(k\text{.}\)
Use the formula for \({n \choose k}\) to give an algebraic proof of the identity.
Give a combinatorial proof of the identity.