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Suppose you own \(x\) fezzes and \(y\) bow ties. Of course, \(x\) and \(y\) are both greater than 1.

  1. How many combinations of fez and bow tie can you make? You can wear only one fez and one bow tie at a time. Explain.

  2. Explain why the answer is also \({x+y \choose 2} - {x \choose 2} - {y \choose 2}\text{.}\) (If this is what you claimed the answer was in part (a), try it again.)

  3. Use your answers to parts (a) and (b) to give a combinatorial proof of the identity

    \begin{equation*} {x+y \choose 2} - {x \choose 2} - {y \choose 2} = xy.\text{.} \end{equation*}
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