Generalize from Binomial to Multinomial

Question

Trying to find a generic formula for the probability of N players out of P players picking the same value from a dice with D sides.

i.e. If we have 3 players with 3 sided dice, you have 3/27 probability of them getting the same number (1, 1, 1), (2, 2, 2), (3, 3, 3). You also have 18/27 probability of 2 of them getting the same number.

I was able to generalize for a 2 sided-dice (coin-flip).

Variables Summary:

1. Dice faces: D
2. Players: P
3. Needed number of players with same side: N

$$ (D)*({P\choose N}/(D)^{P}) $$

When I tried to apply the above formula for more than 2 sides, it didn't give the correct value.

If I have 5 people who roll (1, 1, 2, 3, 3), this is considered as 1 sample with 2 matches.

Any pointers?

Edit: Fixed wrong variable

Answer 1

(I take it by X you mean N.) The problem is still essentially binomial; for each side of the die, you have the outcomes "yes, that side" and "no, one of the D-1 other sides." The standard binomial formula is

$$ Pr(N) = {P\choose N}q^N (1-q)^{(P-N)} $$

For your case, $q=1/D$, yielding

$$ {P\choose N}(\frac{1}{D})^N (1-\frac{1}{D})^{(P-N)} $$

Then multiply by D (as you did in your formula) to account for the D distinct sides:

$$ D{P\choose N}(\frac{1}{D})^N (1-\frac{1}{D})^{(P-N)} $$