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Chapter 20 Combinatorial Proof
Exercises Practice Problems
1.
Give a combinatorial proof for the identity
\begin{equation*}
{3n \choose 3} = 3 {n \choose 3} + 6n {n \choose 2} + n^3.
\end{equation*}
2.
Give a combinatorial proof for the identity
\begin{equation*}
\sum_{k=0}^n \left( {n \choose k} \right)^2 = { 2n \choose n}.
\end{equation*}
Hint: you will need the identity \({r \choose s} = {r \choose r-s}\text{.}\)
3.
Give a combinatorial proof for the identity
\begin{equation*}
{n + m \choose r} =
\sum_{i=0}^r {n \choose i} {m \choose r-i}.
\end{equation*}
4.
Give a combinatorial proof for the identity
\begin{equation*}
{k \choose k} + {k+1 \choose k} + {k+2 \choose k} + \cdots + {n \choose k} = {n+1 \choose k+1}.
\end{equation*}
Hint: For the LHS, we have split into cases according to the LARGEST ELEMENT in our \((k+1)\)-set of \([n+1]\text{.}\)