Probability of throwing n different numbers in m throws of a dice

#1
Folks

I know how to calculate the probability of throwing the same number in every throw of a dice (die, if you prefer!), and I know how to calculate the probability of throwing a different number in each throw of a dice. But how do I calculate the probability of throwing n different numbers in m throws of a dice?

Just to be clear, it doesn't matter what the numbers are or how many times each number is thrown – so two ones and three twos in five throws will be equivalent to four threes and one four in the same number of throws, for example.

Thanks

Remster
 
#2
I don't know of a single formula, and my guess would be that there isn't one. However, the probabilities are easy to calculate using an iterative formula. If p(n,m) is the probability of getting exactly n different numbers after m throws, then just before the last throw there are either n numbers and you get another of the ones you already have, or there are n-1 numbers and you get a different one. So p(n,m) = p(n,m-1)*n/6 + p(n-1,m-1)*(6-(n-1))/6.
To start off, p(0,m) = 0, p(n,0) = 0 and p(1,1) = 1. Set it up in Excel, put in the formula, and copy in all directions. Post back if you need to see an actual sheet. kat