Ten unit steps from 0 ending at 4

Count the sequences $x_0,x_1,\dots,x_{10}$ of numbers such that $x_0=0$, $x_{10}=4$, and each consecutive pair differs by exactly one, that is $|x_i-x_{i-1}|=1$ for $i=1,\dots,10$.

$120$. Each step is $+1$ or $-1$; to net $+4$ in $10$ steps you need $7$ up-steps and $3$ down-steps. Choosing where the down-steps go gives $\binom{10}{3}=120$.