(Probability Question)
I have a number of $N$ types of gems that I have handed out to $M$ people, using some random probability function for each person. Thus: $ \begin{align} P_1^1 + P_2^1 + P_3^1 + P_4^1 + \dotsb + P_n^1 &= 1 \\ P_1^2 + P_2^2 + P_3^2 + P_4^2 + \dotsb + P_n^2 &= 1 \\ P_1^3 + P_2^3 + P_3^3 + P_4^3 + \dotsb + P_n^3 &= 1 \\ P_1^4 + P_2^4 + P_3^4 + P_4^4 + \dotsb + P_n^4 &= 1 \\ P_1^5 + P_2^5 + P_3^5 + P_4^5 + \dotsb + P_n^5 &= 1 \\ \vdots & \\ P_1^m + P_2^m + P_3^m + P_4^m + \dotsb + P_n^m &= 1 \end{align} $
Thus, each person has a type of gem with a certain probability (each person can only have one type of gem).
An example for $N = 3$ , $M = 3$ would be (each row is a different person, each column a gem of type different type):
$0.3 ~0.5~ 0.2$
$0.8 ~0.1~ 0.1$
$0.4 ~0.3~ 0.3$
Is there a way of finding out the probability of at least one person having a gem of type $n$, at least one person having a gem of type $n-1$, but none having one of $n$, at least one person having a gem of type $n-2$, but none having a gem of type $(n-1)$ and $n$ etc., without enumerating all the joint probabilities ($N^M$ possible combinations) ?