# Proof Involving Binomial Theorem

• Feb 27th 2010, 02:12 PM
nauticaricky
Proof Involving Binomial Theorem
I have spent over 5 hours now trying to figure out how to prove this, and I still have got nothing. My teacher claims there is a somewhat simple way to prove this, and I cannot find it.

To see the thing that I am to prove, click this link : Yfrog - prooffm - Uploaded by Imageshack user

I am quite certain, but not entirely sure, that I have to use mathematical induction to prove it holds for n=0 and for n=n+1, but besides that, I am not sure how to go about things. I have tried everything I can think of.

If necessary, here is a link to the electronic version of our text book : http://math.sfsu.edu/beck/papers/aop.pdf . This problem is one from chapter #4. I feel like I should be able to use the Binomial Theorem somehow, but I cant get it.

Help would be appreciated great amounts! I must have this finished on monday.

Richard
• Feb 27th 2010, 02:45 PM
NonCommAlg
Quote:

Originally Posted by nauticaricky
I have spent over 5 hours now trying to figure out how to prove this, and I still have got nothing. My teacher claims there is a somewhat simple way to prove this, and I cannot find it.

To see the thing that I am to prove, click this link : Yfrog - prooffm - Uploaded by Imageshack user

I am quite certain, but not entirely sure, that I have to use mathematical induction to prove it holds for n=0 and for n=n+1, but besides that, I am not sure how to go about things. I have tried everything I can think of.

If necessary, here is a link to the electronic version of our text book : http://math.sfsu.edu/beck/papers/aop.pdf . This problem is one from chapter #4. I feel like I should be able to use the Binomial Theorem somehow, but I cant get it.

Help would be appreciated great amounts! I must have this finished on monday.

Richard

differentiate both sides of the identity $(x+1)^n=\sum_{k=0}^n \binom{n}{k}x^k$ with respect to $x$ and then put $x=1.$
• Feb 27th 2010, 08:33 PM
icefirekid
Hi. Hope this helps. I am using Combinatoric Proof to prove it.

Assume we have a set of n elements.

On the left hand side : We are couning the total number of subsets that can be formed with the n elements and from each of those subset, we are choosing one element. This is like choosing a committee and from the committee we are choosing a leader.

On the right hand side : We first choose the so called leader from the n elements and from the remaining n-1 elemtents, we form the remaining of the subset/committee. Thus having $n2^{n+1}$

Hope it makes sense.