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Prove the identity $\binom{n}{r}\binom{r}{k} = \binom{n}{k}\binom{n−k}{r−k} ,$ whenever $n, r,$ and $k$ are nonnegative integers with $r \leq n$ and $k \leq r,$

  1. using a combinatorial argument.
  2. using an argument based on the formula for the number of $r$-combinations of a set with $n$ elements.
in Combinatory
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