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  Discussion about math, puzzles, games and fun.   Useful symbols: √ ∞ ≠ ≤ ≥ ≈ ⇒ ∈ Δ θ ∴ ∑ ∫ π -

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Topic review (newest first)

cooljackiec
2013-01-02 07:09:54

Algebraically?

anonimnystefy
2013-01-02 06:17:24

Another way:

Write the sum on the left as:



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!

scientia
2013-01-02 05:40:09

You can check that



(Expand LHS and show it's equal to RHS.) Hence







Adding up



Rearrange, noting that
, and you're done.

cooljackiec
2013-01-02 05:06:34

how would we prove the identity:


in algebra?

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