Ytukyg

For any subset $Ssubseteq{1,2,ldots,15}$, call a number $n$ an anchor for $S$ if $n$ and $n+#(S)$ are both elements of $S$. For example, $4$ is an anchor of the set $S={4,7,14}$, since $4in S$ and $4+#(S) = 4+3 = 7in S$.

Given that $S$ is randomly chosen from all $2^{15}$ subsets of ${1,2,ldots,15}$ (with each subset being equally likely), what is the expected value of the number of anchors of $S$?

I don't know how to start this problem.

"...I don't know how to start this problem."

For $ninleft{ 1,2,dots,15right} $ let $a_{n}$ denotes the
number of subsets of $left{ 1,2,dots,15right} $ with the property
that $n$ is an anchor of it.

Further let $X_{n}$ take value $1$ if $n$ is an anchor of the randomly chosen $S$
and let it take value $0$ otherwise.

Then $X:=X_{1}+cdots+X_{15}$ is the number of anchors of $S$ and
with linearity of expectation we find:

$mathbb{E}X=sum_{n=1}^{15}mathbb{E}X_{n}=sum_{n=1}^{15}Pleft(X_{n}=1right)=sum_{n=1}^{15}Pleft(ntext{ is an anchor of }Sright)=2^{-15}sum_{n=1}^{15}a_{n}$

Now it remains to find the $a_{n}$.

Maybe you should first give that a try yourself.