A card game using 36 unique cards: four suits (diamonds, hearts, clubs, and spades) with cards numbered from 1 to 9 in each suit. A hand is a collection of 9 cards, which can be sorted however the player chooses. What is the probability of getting all four of the 1s?
The total number of possible hands is the number of ways to choose 9 cards out of 36. This can be calculated using the combination formula:
C(36, 9) = 36! / (9! * (36-9)!) = 9,075,135
To calculate the probability of getting all four 1s, we need to count the number of hands that contain all four 1s. There are 4 ways to choose which suit the first 1 comes from, then 3 ways to choose which suit the second 1 comes from (since we can't choose the same suit twice), and so on. So the number of hands with all four 1s is:
4 * 3 * 2 * 1 * C(32, 5)
The factor of 4 * 3 * 2 * 1 comes from choosing which suits the four 1s come from. After those four cards are chosen, we need to choose 5 more cards from the remaining 32. This can be done in C(32, 5) ways.
So the probability of getting all four 1s is:
(4 * 3 * 2 * 1 * C(32, 5)) / C(36, 9) = 48/935.
