  We study the random variable B(c,n), which counts the number of balls that
must be thrown into n equally-sized bins in order to obtain c collisions. The
asymptotic expected value of B(1,n) is the well-known $\sqrt{n\pi/2}$ appearing
in the solution to the birthday problem; the limit distribution and asymptotic
moments of B(1,n) are also well known. We calculate the distribution and
moments of B(c,n) asymptotically as n goes to infinity and c = O(n).
  Our main tools are an embedding of the collision process, realizing the
process as a deterministic function of the standard Poisson process, and a
central limit result by Renyi.
