I have a set $S = \lbrace A,B,C,D,E,F,G,H,I,J,K,L\rbrace$, that I want to create a multiset with.
Suppose I create the multiset $MS$ from combination with repetition of 4 from 12 $Cr(N=12, R=4)$ from the set $S$. The members should be like this:
$MS = \lbrace AAAA, AAAB, ..., LLLL\rbrace$ with size $Cr(N=12, R=4) = 1365$
Let say I have a discrete random variable $X = $ the number of distinct element. For example: $AAAA = 1$, $AABC = 3$, $CDEF = 4$, and $AAKK = 2$.
My question is:
- How to calculate the probability of each $P(X=1), P(X=2), ..., P(X=R)$ ..?
I know $P(X=1)$, which is the proportion of only $\lbrace AAAA, BBBB, ..., LLLL\rbrace$ in the multiset $MS$. Given the example, it would be $\frac{N}{Cr(N=12, R=4)} = \frac{12}{1365} = 0.009$
I also know $P(X=R)$ which is the proportion of distinct combination $\lbrace ABCD, ABCE, ..., IJLK\rbrace$ from the multiset $MS$. I could obtain it with combination without repetition like this $\frac{C(N=12, R=4)}{Cr(N=12, R=4)} = \frac{495}{1365} = 0.3626$.
What I don't know is the value in between and the methods to calculate them.
Later if I understand it, I would like to be able to calculate $P(X=1), P(X=2), ..., P(X=r)$ in general way for combination with repetition $Cr(N=n, R=r)$ with large $n$ and $r$ eg. $n>10^6$ and $r>10^3$.