How many sequences satisfy $X_0 = 0$, $X_{10} = 4$ and $|X_{i} - X_{i - 1}| = 1$ for $1 \leq i \leq 10$?
I was doing some research, and I think the answer may involve these: https://en.wikipedia.org/wiki/Catalan_number
But I am not so experienced with them. Maybe someone can help me.
I tried working backwards. For example, we know $X_9$ must be either $3$ or $5$. But it gets pretty messy. Maybe someone can make progress with this.
If there are $k$ "up" moves, there must be $10-k$ "down" moves. From the boundary conditions, $k-(10-k)=4$, so $k=7$. There are $10\choose 7$ ways to distribute $7$ up moves among ten moves in total.