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#1 2013-01-02 05:06:34

cooljackiec
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Proofs

how would we prove the identity:


in algebra?

Last edited by cooljackiec (2013-01-02 05:07:38)


I see you have graph paper.
You must be plotting something
lol

#2 2013-01-02 05:40:09

scientia
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Re: Proofs

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)

#3 2013-01-02 06:17:24

anonimnystefy
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Re: Proofs

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 (2013-01-02 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

#4 2013-01-02 07:09:54

cooljackiec
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Re: Proofs

Algebraically?


I see you have graph paper.
You must be plotting something
lol

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