Counting the Number of Votes by Induction Suppose that $A$ has $r$ votes and $B$ has $u$ votes where $r\ge u\ge 0$.
Show by induction that the number of ways to count $n=r+u$ votes is

$\frac{r-u+1}{r+1}\binom{r+u}{r}$.

I verified the case for $n=1$ by taking $r=1,u=0$ but I can't prove the general case.
How may I show this using induction. Please help.

In equation $(10)$ of this answer, it is shown that
$$
b_{s,n}=\frac{s+1}{\frac{n+s}2+1}\binom{n}{\frac{n+s}2}\,[2\mid n+s]\tag1
$$
where $b_{s,n}$ is the number of strings of $+1$ and $-1$ of length $n$ with non-negative partial sums that total to $s$. However, that answer, and the answer it is based on, use generating functions and induction.
Plugging in $n=r+u$ and $s=r-u$, we get
$$
\frac{r-u+1}{r+1}\binom{r+u}{r}\tag2
$$
