## anonymous 3 years ago I was enumerating the elements of the power set of this set S:= {1,2,3,4,5} and I thought that the number of these elements could be obtained with this: $\#\wp S = 1 + \sum_{k=1}^n {n\choose k}$ where $$n=\#S$$ I saw that it holds for this set. But I'm not sure what if it could be applied to a different kind of set.

1. anonymous

That formula is right, but there is a much shorter formula for the number of elements in the power set of an n-element set. If write out the power sets for sets with 2, 3 and 4 elements, it should become clear what the shorter formula is.

2. anonymous

Yeah I know. The formula is $$2^n$$

3. anonymous

I'm ashamed this is actually the binomial theorem

4. anonymous

the binomial expansion of $$(x+y)^n$$ is $\Large (x+y)^n=\sum_{k=0}^n \left(\begin{matrix}n \\ k\end{matrix}\right)x^ky^{n-k}$ put $$x=y=1$$ and you'll get the answer to your question.

5. anonymous

$\#\wp S=2^{\#S}= \sum_{k=0}^{n} {n\choose k}$

6. anonymous

thank you @sirm3d

7. anonymous

YW