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
since deck contains four suits of 13 cards each:
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
total choices.
However, this count includes the straight flushes. Removing the 40 straight flushes leaves us with 10,200 straights.
iii)
There are
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
hands of two pairs.