SOLUTION: if 5 cards are drawn at random from a deck of 52, what is the probability of pulling 3 cards of one suit and 2 cards of a second suit?

Question 1144484: if 5 cards are drawn at random from a deck of 52, what is the probability of pulling 3 cards of one suit and 2 cards of a second suit?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The number of 5-card hands is the denominator of the probability fraction: 52C5 = 2598960.

Here are two different ways to analyze and solve the problem of finding how many of those hands are full houses.

(1) Choose 1 of the 13 ranks; then choose 3 of the 4 suits of that rank: 13C1 * 4C3 = 13*4 = 52
Then choose 1 of the remaining 12 ranks and then choose 2 of the 4 suits of that rank: 12C1 * 4C2 = 12*6 = 72
Total number of full house hands: 72*52 = 3744

(2) Choose 2 of the 13 ranks; then choose 1 of the 2 ranks to be the one that gets 3 suits: 13C2 * 2C1 = 78*2 = 156
Then choose 3 of the 4 suits of the first rank and 2 of the 4 suits of the second rank: 4C3 * 4C2 = 4*6 = 24
Total number of full house hands: 156*24 = 3744

So the probability of getting a full house is 3744/2598960 = 0.00144 to 5 decimal places
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