# Thread: Evaluating a Series of Binomial Co-efficients

1. ## Evaluating a Series of Binomial Co-efficients

Hey guys,

I've been scratching my head over this one:

Evaluate:
$\displaystyle \binom{40}{0}+\binom{40}{1}+\binom{40}{2}+\dots+\b inom{40}{40}$

That is to say,

$\displaystyle \sum_{n=0}^{40}\binom{40}{n}$

My first thought was to say

$\displaystyle \sum_{n=0}^{40}\binom{40}{n} = \sum_{n=0}^{40}\frac{40!}{40(40-n)!} = \sum_{n=0}^{40}\left(\frac{A}{40}+\frac{B}{(40-n)!}\right)$

Then to find $\displaystyle A$ and $\displaystyle B$ by partial fractions and do a summation table, but couldn't get it out. Any suggestions from anyone?

2. Originally Posted by Stephen Shaw
Evaluate:
$\displaystyle \binom{40}{0}+\binom{40}{1}+\binom{40}{2}+\dots+\b inom{40}{40}$
That is to say,
$\displaystyle \sum_{n=0}^{40}\binom{40}{n}$
Think about it for a moment.
That is simply the number of subsets of a set of 40 elements: $\displaystyle 2^{40}$.
Another way. $\displaystyle \left( {x + y} \right)^{40} = \sum\limits_{k = 0}^{40} {{40 \choose k}x^{40 - k} y^k }$
Let $\displaystyle x=1\;\&\;y=1$ we have $\displaystyle \left( {2} \right)^{40} = \sum\limits_{k = 0}^{40} {{40 \choose k} }$