How many 5card hands have a single pair and no 3ofakind or 4

How many 5-card hands have a single pair and no 3-of-a-kind or 4-c a-kind? How many 5-card hands have exactly a pair of aces?

Solution

Hi, I am Waqar. Please see below for detailed solution and lead to your answer.

(a)

This the hand with the pattern AABCD, where A, B, C and D are from the distinct \"kinds\" of cards: aces, twos, threes, tens, jacks, queens, and kings

=> There are 13 kinds, and four of each kind, in the standard 52 card deck.

=> The number of such hands is (13-choose-1)*(4-choose-2)*(12-choose-3)*[(4-choose-1)]^3.

=> If all hands are equally likely, the probability of a single pair is obtained by dividing by (52-choose-5).

=> This probability is 0.422569.

(b)

This hand has the pattern AABBC where A, B, and C are from distinct kinds.

=>The number of such hands is (13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1).

=>After dividing by (52-choose-5),

=> The probability is 0.047539.

Happy to help you

Happy Chegging