It is one of those problems that invites us to list paths one by one. There is a much more elegant way to solve it.
12-step ladder
Riddle statement
You climb a 12-step ladder. In each movement you can advance 1 or 2 steps.
How many different ways can you get to the top?
Show solution
Solution
Answer: 233 ways.
Let $F(n)$ be the number of ways to reach the step $n$.
The last movement can only be of two types:
or you come from the step $n-1$ with a jump of 1;
or you come from the step $n-2$ with a jump of 2.
Therefore, $ F(n)=F(n-1)+F(n-2). $
The initial conditions are: $ F(1)=1,\qquad F(2)=2. $
From there: $ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ 233. $
So, for 12 steps, $ F(12)=233. $
It is the same Fibonacci recurrence, but displaced by the initial conditions.