A poker hand consists of five cards from a standard deck of 52. (See the chart preceding Example 8.) Find the number of different poker hands of the specified type. HINT [See Example 8.]
Straight (five cards of consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A, not all of the same suit) (Note that the ace counts either as a 1 or as the denomination above king.)
There are 10 different types of straights or straight flushes:
1. A,2,3,4,5
2. 2,3,4,5,6
3. 3,4,5,6,7
4. 4,5,6,7,8
5. 5,6,7,8,9
6. 6,7,8,9,10
7. 7,8,9,10,J
8. 8,9,10,J,Q
9. 9,10,J,Q,K
10. 10,J,Q,K,A
We can choose the type of straight or straight flush from
the above list 10 ways.
We can choose the suit for the lowest card 4 ways.
We can choose the suit for the next to lowest card 4 ways.
We can choose the suit for the middle card 4 ways.
We can choose the suit for the next to highest card 4 ways.
We can choose the suit for the highest card 4 ways.
That is 10×4×4×4×4×4 = 10×45 = 10240 straights or straight flushes.
But we must subtract from that the number of straight flushes,
where they are all of the same suit.
So we calculate the number of straight flushes:
There are 10 types of straight flushes.
There are 4 suits each type can have.
So there are 10×4 or 40 possible straight flushes.
So there are 10240 - 40 or 10200 possible straights.
Edwin