A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A).
A)1,287 hands
B) 5,108 hands
C) 792 hands
D) 5,148 hands
E) 1,247 hands

Question

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A). ​
A)1,287 hands
B) 5,108 hands
C) 792 hands
D) 5,148 hands
E) 1,247 hands

Quizplus · Accepted Answer

To calculate the number of different flush hands (excluding straight flushes and royal flushes), we first choose a suit (4 ways), and then choose 5 cards out of the 13 available in that suit. The number of ways to choose 5 cards from 13 is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items, $k$ is the number of items to choose, and "!" denotes factorial, the product of all positive integers up to that number. Thus, the calculation is $4 \times C(13, 5) = 4 \times \frac{13!}{5!(13-5)!} = 4 \times \frac{13!}{5!8!} = 4 \times 1,287 = 5,148$.

A Poker Hand Consists of Five Cards from a Standard