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**cooljackiec****Member**- Registered: 2012-12-13
- Posts: 178

how would we prove the identity:

in algebra?

*Last edited by cooljackiec (2013-01-01 06:07:38)*

I see you have graph paper.

You must be plotting something

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**scientia****Banned**- Registered: 2009-11-13
- Posts: 224

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 (2013-01-01 07:06:36)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

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!

*Last edited by anonimnystefy (2013-01-01 07:19:47)*

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**cooljackiec****Member**- Registered: 2012-12-13
- Posts: 178

Algebraically?

I see you have graph paper.

You must be plotting something

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