Math Is Fun Forum

cooljackiec

Member

Registered: 2012-12-13

Posts: 186

Proofs

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

scientia

Member

Registered: 2009-11-13

Posts: 224

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

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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 (2013-01-01 07:19:47)

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

cooljackiec

Member

Registered: 2012-12-13

Posts: 186

Re: Proofs

Algebraically?

---

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