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# Probability

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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?

Mar 22, 2023

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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.

Mar 22, 2023