Puzzle 3 of 26 · Medium

Dealt a straight

Your answer

A straight is 5 cards of consecutive ranks, like 5-6-7-8-9 (the ace can be low, A-2-3-4-5, or high, 10-J-Q-K-A). Of the 2,598,960 possible 5-card hands, how many are a straight but not all one suit?

A straight: 5-6-7-8-9 in mixed suits.

Right answers go through as you type.

Show the solution

Answer: 10,200

Ranks

\displaystyle 10

Suits

\displaystyle 4^{5}

\displaystyle 10{,}240

Step 1 · Choose the lowest card’s rank
$\displaystyle A,\, 2,\, 3,\, \dots \text{ up to } 10 = 10$
The lowest rank fixes the other four. It can be anything from ace (A-2-3-4-5) up to 10 (10-J-Q-K-A).
Step 2 · Choose a suit for each card
$\displaystyle 4^{5} = 1{,}024$
Each of the 5 cards can be any of the 4 suits.
Step 3 · Take off the ones all in one suit
$\displaystyle 10 \times 1{,}024 - 10 \times 4 = 10{,}200$
10 $\text{ranks } \times 4$ $\text{suits } = 40$ hands are straight flushes, which don’t count. That leaves a chance of about 0.39%.

Answer $\displaystyle 10{,}200$

Step 3 of 3

The method · Unit 1, CountingCombinationsCount groups when order doesn’t matter

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