Question

In the game of poker, determine the number of ways a straight (five cards with consecutive values, such as A 2 3 4 5 or 7 8 9 10 J or 10 J Q K A, but not all of the same suit) can be picked.

Solution

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We notice that there are 4 categories with 13 cards ( 52 cards ). Therefore, there are (525)\binom{52}{5} different ways to choose 5 cards. Now,we require that the card values be consecutive with at least two cards of a different category. If we neglect the category, there are 13 different ways to chose 5 consecutive cards

(A,2,3,4,5),(2,3,4,5,6),,(K,A,2,3,4)\begin{equation} (A, 2, 3, 4, 5),(2,3,4,5,6), \ldots,(K, A, 2,3,4) \end{equation}

First, we assign values to the first 3 cards. There are 13 choices for the cards, and these cards can be drawn in (43)\binom{4}{3} different ways.

On the other hand, there are 12 choices for the remaining two cards and (42)\binom{4}{2} ways to draw these cards.

Therefore, we have

[13(43)][12(42)]=13(4)(12)(2)=3744possibilities\begin{equation} \left[13\binom{4}{3}\right]\left[12\binom{4}{2}\right]=13(4)(12)(2)=3744\quad\text{possibilities} \end{equation}

The probability result

P=3744(525)=37442598960=64165=0.00144058\begin{equation} P=\frac{3744}{\binom{52}{5}}=\frac{3744}{2598960}=\frac{6}{4165}=0.00144058 \end{equation}

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