I want to describe the distribution where different coloured balls are drawn from a bag with replacement (so far, I know this is the multinomial). However, the observer only knows that he got different coloured balls, he can't say which colour they were specifically.
So if there were red, blue, and green balls in the bag and he reports a sampling of (3, 2, 1) then he could have sampled (3 reds, 2 blue, 1 green) or (3 blue, 2 red, 1 green) or (3 green, 2 blue, 1 red) or ...
I need to calculate the probability of observing a specific numerical vector from this distribution, given the composition of balls in the bag. At the moment I'm treating the samples as unordered, so P(X=(1,2,3) == P(X=(3,2,1)), and summing the individual multinomial probabilities over the permutations of X. Is this correct?
I feel like this distribution (and the without replacement sampling version of it) must be described somewhere in the literature but I don't know where to look. I'm also curious about the 'ordered' version, where reporting X = (2,3,1) implies that the colour sampled 2 times was the first colour found in the sampling.