Summing Factorials: Solving the Homework Statement

[dystplan]

Homework Statement

\sum_{n=0}^{100} 1/n!(100-n)!

The Attempt at a Solution

Other then obvious attempts to make sense of the equation's incremental and decrements divisor, I can't figure out where to start with this question. Some assistance would be greatly appreciated.

 

[Dick]

It looks like it's closely related to the sum of binomial coefficients. What's the sum of C(100,n) for n from 0 to 100? Is that enough of a hint?

 

[dystplan]

C(100,n) being = 100!/n!(100-n)! ? hmmm, unfortunately I don't see where that's going =/

Man this one is throwing me for a loop, only question I haven't managed and due tomorrow.

 

[Dick]

C(100,n) being = 100!/n!(100-n)! ? hmmm, unfortunately I don't see where that's going =/

Man this one is throwing me for a loop, only question I haven't managed and due tomorrow.

Yes, that's C(100,n). There is a simple formula for the sum of the binomial coefficients. It's related to the value of (1+1)^100. Don't know it? Expand (1+1)^100 using the binomial theorem.

 

[dystplan]

well; thanks. But I'm still just totally lost. (1+1)^100 is a massive equation when expanded.

 

[dystplan]

Unless... x/y = 0 or 1 in the binomial theorem? That would make it easy.

Makes it = 1/100! ?

 

[Dick]

well; thanks. But I'm still just totally lost. (1+1)^100 is a massive equation when expanded.

Oh come on, (1+1)^100=2^100. That's an easy enough number to write down. Now what does that have to do with the sum of the binomial coefficients C(100,n)?? C(100,0)+C(100,1)+...+C(100,100). I'm not asking you to evaluate each one. Just think about it.

 

[dystplan]

wait;

2^100/100!

 

[dystplan]

wait;

2^100/100!

 

[Dick]

wait;

2^100/100!

You aren't just guessing, I hope.