Does this probability distribution have a name? We have a set of numbers, of size $m$. We are going to pick $a$ numbers with uniform probability from that set, with replacement. Let X be the random variable denoting the probability of having X of those picks distinct (exactly X distinct values are picked).
Motivation: I need to calculate this probability in order to calculate a more advanced distribution regarding Bloom filters, in particular the distribution of the number of bits set to 1 in a Bloom filter. 
Letting that aside, I am having trouble formulating the the PMF for X. I've tried to look out for multi-variate binomial distribution but I couldn't relate it to what I want to do.
The question is whether there is such a probability distribution in the literature, and if now, how can I approach this problem ?
Thanks.
Update:
I have managed to make a formulation: the probability we pick $x$ distinct values is 
$$
\frac{1}{m} \frac{1}{m-1} \cdots \frac{1}{m-x+1}
$$
And the probability of picking the rest of our $a-x$ picks in that set of $x$ values is
$$
\left(\frac{x}{m}\right)^{a-x}
$$
Finally, the number of such configurations is $\binom{m}{x}$. Multiplying all that together and simplifying gives us a PMF
$$
P(X=x;a,m) = \frac{ \left( \frac{m}{x} \right) ^{x-a}}{x!}
$$
Does that seem to make any sense ?

The number of multisets from a set of size $m$ with cardinality $a$ is $\binom{m+a-1}{a}$.
