jmc

$\left(\genfrac{}{}{0ex}{}{n}{2}\right)=\frac{n(n-1)}{2}$. In order for this fraction to be odd, neither $n$ nor $n-1$ can be divisible by $4$, because only one of $n$ and $n-1$ can be even. There are $25$ integers where $n$ is divisible by $4$, namely the multiples of $4$ from $4$ to $100$. There are $24$ integers where $n-1$ is divisible by $4$. We can obtain these integers by incrementing all the multiples of $4$ by $1$, but we must not include $100$ since $100+1=101>100$. Therefore, there are $49$ invalid integers, so there are $99-49=\boxed{50}$ valid integers.