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 $\binom{52}{5}$ 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

$$$(A, 2, 3, 4, 5),(2,3,4,5,6), \ldots,(K, A, 2,3,4)$$$

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

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

Therefore, we have

$$$\left[13\binom{4}{3}\right]\left[12\binom{4}{2}\right]=13(4)(12)(2)=3744\quad\text{possibilities}$$$

The probability result

$$$P=\frac{3744}{\binom{52}{5}}=\frac{3744}{2598960}=\frac{6}{4165}=0.00144058$$$

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