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Let $0 \leq r \leq n$. Can someone give an intuitive explanation for why the following identity holds or give me a hint on proving it?

$\dbinom{n}{n-r} = \dbinom{n}{r}$

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up vote 12 down vote accepted

One way to pick $3$ among $10$ apples, is to pick the $7$ that you will not keep.

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Very nice. Thank you! – Student Feb 5 '12 at 1:50

$\binom{n}r$ is the number of ways to choose $r$ objects from a set of $n$ objects. When you choose $r$ objects, you’re rejecting the other $n-r$ objects, so $\binom{n}r$ is also the number of ways to pick $n-r$ of the $n$ objects to be rejected. But that’s clearly the same as the number of ways to choose $n-r$ of the $n$ objects, which is $\binom{n-r}r$: choosing in order to keep and choosing in order to reject are both just choosing.

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