How many different ways can I get up a flight (of stairs) with 11 steps? You can climb either $1$ or $2$ stairs at a time, at any given time. How many ways can you get up $11$ stairs?
I've tried using different cases to solve this. 
So I did:
Case 1: All $1$ steps --> $11  $steps --> number of combinations = $1$ 
Case 2: $9 $one steps, $1$ two step --> number of combinations = $\binom{n}{2} $??
Case 3: $7$ one steps, $2$ two steps --> number of combinations? 
I don't understand how to proceed further. Is there any easier way to do this?

Hint: Start at the beginning, not the end.
Step $1$: If there is one step, there is only one way to take the stairs.
Step $2$: If there are two steps, you can either take take $2$ steps or one step twice, leading to two ways to take the stairs.
Step $3$: Consider your first step, if you start with $1$ step, then $2$ steps remain and we know that there are two ways to go up two stairs.  On the other hand, if you start with $2$ steps, this leaves one step and we know that there is one way to go up one stair.  This gives $1+2=3$ ways to go up three steps.
Step $4$: Consider the first step, if you start with $1$ step, then $3$ steps remain and we know that there are three ways to go up three stairs.  On the other hand, if you start with $2$ steps, then there are only two steps left, and we know that there are $2$ ways to go up $2$ steps.  This gives $2+3=5$ ways to go up $4$ steps.
Continue in this way.  You may recognize the numbers that you get as being in a popular sequence.
