How many cycles, $C_{4}$, does the graph $Q_{n}$ contain? Let $n \geq 2$ be an integer. How many cycles of length $4$ (that is, subgraphs isomorphic to $C_4$) does the hypercube graph $Q_n$ contain?


Examples: For $n = 2$, the hypercube graph $Q_2$ is a single $4$-cycle, so the answer in this case is $1$. For $n = 3$, the hypercube graph $Q_3$ is a cube, and the answer is $6$.

I was told that $n$-dimensional cube graphs, $Q_{n}$ can be represented by binary but how can that help me approach this question?
How many $C_4$s pass through a given vertex $v$? There are  two edges in this cycle through
$v$. There are $n$ edges through $v$ so $\binom {n}2$
pairs of edges. Once we have these two edges there is
just one way to complete the $C_4$. Don't forget there are $2^n$
vertices overall and each $C_4$ features four of them.
Here is another way to look at it: $Q_n$ is a two-dimensional projection of an $n$-dimensional hypercube, so each distinct $C_4$ in $Q_n$ is a face of the $n$-dimensional hypercube. 

When $n\ge 3$ the $n$-dimensional cube satisfies the generalized Euler polytope formula
$$
V-E+F=1+(-1)^{n-1}.
$$
For $Q(n)$ we have $V=2^n$ and $E=n2^{n-1}$ so that
$$
F=n2^{n-1}-2^n+1+(-1)^{n-1}.
$$

As to how $Q_n$ is represented with binary strings: Each vertex in $Q_n$ corresponds with each of the possible $n+1$-digit binary string and an edge exists between two vertices if their corresponding strings differ in exactly one digit.