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?

One can define the $n$-th Fibonacci number (*) as the number of ways you can write $n$ as a sum of $1$'s and $2$'s. 
Thus, your answer is the $11$-th Fibonacci number.
(*) Note that there are several definitions of the Fibonacci numbers in common use, so make sure you use the right one, which matches with this definition (note that $F_1 = 1, F_2 = 2$). 
