Heres the scenario: Let's say we get dealt 17 cards and want to know the probability of getting a flush (five cards of the same suit) that can only be spades. No other suit matters. We know that one spade in the deck was taken out. So that leaves us with 12 spades left in a 51 card deck. How would we determine the probability of getting five spades if dealt 17 cards?
I'm pretty sure math stack exchange would be more helpful but to calculate this mathematically you should use combinatorics.
The total amount of ways to be dealt 17 cards is 51 Choose 17 (which is around 1.47 x 10^13 ways)
Then we determine how many ways to get a flush (I'm going to assume exactly 5 spades and not more than 5). This is equal to (12 choose 5) * (39 choose 12). The 12 C 5 part accounts for the 5 spades and the 39 C 12 part accounts for the remaining 12 cards that are not spades. (39 = 51 - 12) This value is around 3.09 x 10^12 ways.
The probability is around 20.96%