1
$\begingroup$

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?

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

$\endgroup$
17
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Very nice. Thank you! $\endgroup$ – Student Feb 5 '12 at 1:50
5
$\begingroup$

$\binom{n}r$ 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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.