Question

A flush in a five-card poker hand is five cards of the same suit. The suits are spades, clubs, diamonds and hearts. How many spade flushes are possible in a 52-card deck? How many flushes are possible in any suit?

Answer 1

Answer:

1287 spade flushes possible in a 52-card deck.

1287 flushes in any suit.

Explanation:

We have been given that a flush in a five-card poker hand is five cards of the same suit. The suits are spades, clubs, diamonds and hearts.

To find the number of possible spade flushes in a 52-card deck, we need to find choosing 5 cards from 13 cards as a standard deck has 13 spades altogether.

$^{13}C_5=\frac{13!}{5!(13-5)!}$

$^{13}C_5=\frac{13!}{5!(8)!}\\\\^{13}C_5=\frac{13*12*11*10*9*8!}{5*4*3*2*1(8)!}$

$^{13}C_5=13*11*9\\\\^{13}C_5=1287$

Therefore, there are 1287 spade flushes possible in a 52-card deck.

Since there 1287 possible suits in spade flushes and flushes of different suits are mutually exclusive events, therefore, there will be 1287 flushes in any suit.