Possible ways to get to a point (on a prism)

Question

I could find the answer by trying every possible way. But now i want to know if there is any sort of equation or formula i can use to solve this.

Answer 1

Use variables such as $x$ (down), $y$ (right), $z$ (forward) for one step in each direction. Here the sequence would be $$x,y,y,z,z$$

Now find no of ways $N$ to rearrange this sequence. Here that would be $$N = \dfrac{5!}{2!\cdot 2!} = 30$$

Now subtract the no. of cases $n$ when $x$ comes third in sequence and z and y come left to it, as Christian Blatter pointed out! That would be $$n=2$$