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In this question I was helped with calculating the probability of drawing specific set of M coloured balls from a set of N coloured balls.

Now I am looking for a solution for an extended problem: what is the probability of finding my specific set of M balls among Q balls (where Q > M), drawn from the same pool of N balls.

For example: there are 100 balls in the box: 50 red, 30 blue and 20 white. I randomly draw (without replacement) 15 balls from the box. What is the probability of finding among them a set of 6 balls, where 2 of them are red, 2 blue and 2 white?

Using the variables:

N = 100: n1 = 50, n2 = 30, n3 = 20

M = 6: m1 = 2, m2 = 2, m3 = 2

Q = 15

While there are methods how to calculate the result for any specific input values, I am looking for a universal algorithm that would give result for any input values. Right now, I find it hard how to make a jump from the simpler problem of drawing exactly M balls: while it feels there should be some kind of multiplier that corrects for the extra freedom of drawing additional Q-M balls, this surely is not simple one, because these extra balls can themselves contain the set M, or there is partial overlapping, so there should be some smart way how to deduct all those duplicate combinations.

Edit: complemented with empirical data from virtual experiment.

I went forward to build a program to perform virtual experiments. Of course, computer random generators are not true random, but that's the best I have and for the sake of this experiment, I think, they are random enough.

So, I created the pool: a set of 2 red, 3 blue and 4 white balls. My sample size is 6 balls, and I consider the sampling a success, if it contains (at least) 1 red, 1 blue and 1 white ball. Running the simulation one million times I get probability of 90.4858%. That does not necessarily exactly matches the calculated probability (whatever it is), but should be pretty close. So, I'd expect that the formula gives such result, given the input parameters.

What you need is called hypergeometric probability. There are totally $binom{100}{15}$ of selecting 15 balls out of 100. You want 6 balls divided into three subsets with specific (color) properties and the rest (15-6) you don't care what properties they have. Since you sample without replacement and experiments are independent you get
$$
P(X)= frac{binom{50}{2} cdot binom{30}{2} cdot binom{20}{2} cdotbinom{100-6}{15-6}}{binom{100}{15}}
$$