How to find the probability that among 5 balls drawn from a box with 20 balls, of which n are white and the rest are black, exactly 2 are white?

Question

Combinatorics

There are 20 balls in a box, n of them are white and the rest are black. 5 balls are drawn at random (n > 5). What is the probability that exactly 2 of them are white? If you try to calculate directly through events, you get very long calculations, there should be an easier way. How to generalize this problem to "M balls, N white, X in the sample, among them Y are white"? (You can consider M > N > X > Y)

Answer 1

to solve the problem, you need to divide the number of options for drawing 2
white and 3 black balls by the number of all options for drawing 5 balls out of 20. since there are 20-n combinations of 3, the total number of ways to draw 5 balls out of 20 is equal to the number of combinations of 20-5.