If you want a less arduous method, you can use a Poisson approximation.
Then you only need p (the probability of making the hand) and y (the number of hands) and you can instantly calculate the (approximate) probability of getting the hand x times.
You can read about it here:
http://en.wikipedia.org/wiki/Poisson_distribution
By the way, the main formula here is
(e^(-yp))((yp)^x)/x!
which can be calculated relatively quickly for most x.