Solution to In the game of poker, five cards are drawn from a deck of 52 cards. … - Sikademy
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Archangel Macsika

In the game of poker, five cards are drawn from a deck of 52 cards. A set of 5 cards is said to be a hand. [A standard 52-card deck contains four suits of 13 cards each. The suits are spades (♠), clubs (♣), hearts (♥) and diamonds(♦). The 13 cards in each suit are labeled A,2,3,. . . ,10,J,Q,K.] (i) How many hands contain Four Of a Kind, i.e. four cards with the same value, four eg: four As, four 7s etc. (ii) How many hands form a Straight, i.e. 5 cards in increasing order such that not all are from the same suit? The lowest card is A although a Straight can also end in an A. [Eg: A♣, 2♠,3♠,4♥,5♦ and 10♦, J♠,Q♥,K♥,A♠ ] (iii) How many hands have Two Pairs, i.e. two cards with one value, two cards with another value and a fifth with a different value? [Eg: 2♠, 2♦, J♣, J♠, 5♥]

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i)

number of hands for one Four Of a Kind is

52-4=48

since deck contains four suits of 13 cards each:

n=13\cdot48=624


ii)

The ranks of the cards in a straight have the form x,x+1,x+2,x+3,x+4, where x can be any of 10 ranks. There are then 4 choices for each card of the given ranks. This yields

4^5\cdot 10=10240   total choices.

However, this count includes the straight flushes. Removing the 40 straight flushes leaves us with 10,200 straights.


iii)

 There are

C^{13}_2=78 choices

for the two ranks of the pairs. There are 6 choices for each of the pairs, and there are 44 choices for the remaining card. This produces

6^2\cdot44\cdot78=123552 hands of two pairs.

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