HMMT 2015 U5

Let $\mathcal{H}$ be the unit hypercube of dimension 4 with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \in$ $\{0,1\}$. (Note that $\mathcal{H}$ has $2^{4}=16$ vertices.) A bug starts at the vertex $(0,0,0,0)$. In how many ways can the bug move to $(1,1,1,1)$ (the opposite corner of $\mathcal{H})$ by taking exactly 4 steps along the edges of $\mathcal{H}$ ?