Expected Number of Birthday-Collision Pairs

In a group of $n$ people whose birthdays are independent and uniform on $\{1,\ldots,365\}$, let $X$ be the number of unordered pairs $(i,j)$ with $i < j$ who share a birthday. Using indicator random variables, find $E[X]$. Then determine the smallest $n$ for which $E[X] \ge 1$.