Combinatorics approach to staircase problem
Let $W(n)$ be the number of ways you can reach step $n$ by 1- or 2-step moves. Then $W(1)=1, W(2)=2$ (by two 1-step moves or one 2-step move). Then to reach $n$ height, your last move is either a 1-step move from height $n-1$ or a 2-step move from height $n-2$, giving us the recurrence $W(n) = W(n-1) + W(n-2)$. I'll leave the solution of this recurrence to you. (Hint: it should be familiar.)