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Topic review (newest first)
We can notice that the number on the RHS is the number of ways to choose r+1 numbers out of the set
The sum on the LHS can be interpreted like this:
If we know that the greatest number we will choose is r+k (for ), then we can choose the rest of the numbers in ways. If we sum all those values for all different values of k between 1 and n-r+1, we will get the total number of ways to choose r+1 numbers from the set mentioned before.
But, we also know that that will be the sum on the RHS, so the RHS and the LHS must be equal!
You can check that
(Expand LHS and show it's equal to RHS.) Hence
Adding up …
Rearrange, noting that , and you're done.
how would we prove the identity: