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Consider the identity:

\begin{equation*} k{n\choose k} = n{n-1 \choose k-1}\text{.} \end{equation*}

  1. Is this true? Try it for a few values of \(n\) and \(k\text{.}\)

  2. Use the formula for \({n \choose k}\) to give an algebraic proof of the identity.

  3. Give a combinatorial proof of the identity.

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