Thread: Binomial Theorem and Sum Conversion

Hi,

I have been asked to derive a simple closed form expression for the
following sum:

n
∑ (-1)^(m+i) *(nCi)*(iCm) = S n, m
i = m

[with m = 0,1,2,....,n ; n= 0,1,2,....] and C is the combinations symbol.


I know that we are meant to be using the binomial theorem but I dont quite know where to go from there. Any help would be much appreciated.

Originally Posted by Number Cruncher 20

Hi,

I have been asked to derive a simple closed form expression for the
following sum:

n
∑ (-1)^(m+i) *(nCi)*(iCm) = S n, m
i = m

[with m = 0,1,2,....,n ; n= 0,1,2,....] and C is the combinations symbol.


I know that we are meant to be using the binomial theorem but I dont quite know where to go from there. Any help would be much appreciated.

Three hints:

1. Prove that is equal to zero if and is equal to 1 if .

2. Prove that for .

3. Deduce from 1. and 2. that your expression is equal to zero if and is equal to 1 if .

Yes I see now, thanks a lot for the help!