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You can check that
(Expand LHS and show it's equal to RHS.) Hence
Adding up …
Rearrange, noting that , and you're done.
Last edited by scientia (2013-01-02 06:06:36)
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!
Last edited by anonimnystefy (2013-01-02 06:19:47)
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment