If $S_n$ is a random walk s.t. $S_0=1$. $S_n=X_1+X_2+...+X_n$ for $n\ge 1$ and for any $i\in N$ $P[X_i=1]=P[X_i=-1]=1/2$
for $r\ge 1$ calculate the number of paths from time $0$ to $2n-1$ s.t., $S_0=1$, $S_{2n-1}=2r$ and the path touches or crosses the abscissa at sometime between $0$ and $2n-1$.
I think it is not required that the path touches the axis only once. So I have to just apply the reflection principle.
Call $p$: moves upwards and to the right and $q$: moves downwards and to the right.
Then $2n-1=p+q$ and $2r=p-q-1$, from which it follows $p=n+r$ and $q=n-r-1$. Hence the number of possible paths $N_{2n-1,2r+1}=\binom{2n-1}{n+r}$.
Is that all ?