Combinatorics approach to staircase problem There are $n$ stairs and a person standing at the bottom wants to reach the top. The person can climb either 1 stair or 2 stairs every time. What is the total number of ways they can reach the top?

There are many ways to do this by using code. But when I first read this problem was to use a combinatorics approach. This problems seems very similar to a "stars and bars" problem. The stars would be the number of stairs and the bars would be the number of steps that be taken.

Given this is the case, my solution is simply: ${(n)+(2-1)}\choose{1}$. 

Is the concept of stars and bars the simplest approach to solving this problem or is there a simpler combinatorics concept at play here and if so, how do I recognize such things? By simple, I mean can I solve this problem by simpler computation than combinatorics (even if it requires some sort of brute force method)?
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.)