Question

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♥]

Answer 1

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.