I am unsure of how to do this problem.
Define a random graph $G=(V,E)$ as follows. The vertex set of $G$ is $V=\left\{1,...,n\right\}$. Now for each pair $\left\{i,j\right\}$ with $i ≠ j ∈ [n]$, add the pair $\left\{i,j\right\}$ to $E$ with probability $\frac{1}{2}$ independent of the choice for every other pair. Let $X_5$ be the $\#$ of independent sets of size $5$ in the graph $G$. Compute the expectation of $X_5$. (Note: an independent set is a set of vertices $I$ in $G$ such that no two vertices in $I$ have an edge between them).
Also, for future reference, what is a good way of approaching these expectational problems? Anyway, my theory on how to solve this problem was to compute the expected value of getting just 1 independent set of size $5$ and then summing it up for all n nodes or something like that. The problem is, I'm not sure how to calculate the probability of getting an independent set either and what to do afterwards.
Can't comment because of reputation.
A good way to tackle expectation problems is to keep in mind that expectation is linear, i.e. $$\sum_i E[X_i] = E\big[\sum_i X_i\big]$$ for any random variables $X_i$, no independence needed.
So as long as you if you have identical subproblems that you can solve, you're done.