TITLE: N unlabelled balls in M labeled buckets
QUESTION [17 upvotes]: So the question is how many ways are there to put N unlabelled balls in M labeled buckets.
One way (I think) I figured it out is $M^N$ ways to put each ball, divided by $N!$ permutations of ball ordering we don't care about:
$$M^N \over N!$$
First, is this correct?
Second, is there another way to approach it without using division?  I need to calculate it in a non-prime modulo space so division isn't available.  I really need a recurrence.

REPLY [3 votes]: The Stars and Bars approach given in the answer by Brian M. Scott can be carried out without division. For example, Pascal's Identity
$$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$$
can be used as a recurrence to calculate the required binomial coefficient.  For large numbers, the procedure is unfortunately quite inefficient: simple operations, but too many of them. 
There is a substantial literature about methods for computing binomial coefficients. For example, Granville gives a sophisticated method or computing binomial coefficents modulo a prime power.