Cardinal of P(S) (Adam's question at Yahoo Answers)

[Fernando Revilla]

Here is the question:

Show for any n ϵ ℤ+ with |S| = n, then |P(S)| = 2n.?

Here is a link to the question:

Show for any n

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Adam,

The number of subsets of $S$ with $k$ elements is $\displaystyle\binom{n}{k}$. So, using the binomial theorem: $$\begin{aligned}\left|\mathcal{P}(S)\right|&= \displaystyle\binom{n}{0}+\displaystyle\binom{n}{1}+\displaystyle\binom{n}{2}+\ldots+ \displaystyle\binom{n}{n}\\&=\displaystyle\binom{n}{0}1^n\cdot 1^0+\displaystyle\binom{n}{1}1^{n-1}\cdot1+\displaystyle\binom{n}{2}1^{n-2}\cdot1^2+\ldots+\displaystyle\binom{n}{n}1^0 \cdot 1^n\\&=(1+1)^n\\&=2^n\end{aligned}$$