Question

a hand consists of 3 cards from a well - shuffled deck of 52 cards. a. find the total number of possible 3 - card poker hands. b. a spade flush is a 3 - card hand consisting of all spade cards. find the number of possible spade flushes. c. find the probability of being dealt a spade flush. click the icon to view a description of a standard deck of playing cards. a. there are a total of poker hands.

Explanation:

Step1: Calculate total number of 3 - card hands

The number of combinations of choosing $r = 3$ cards from $n=52$ cards is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$. So, $C(52,3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100$.

Step2: Calculate number of 3 - card spade hands

There are 13 spade cards in a deck. The number of combinations of choosing 3 spade cards from 13 spade cards is $C(13,3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286$.

Step3: Calculate probability of a spade flush

The probability $P$ of getting a 3 - card spade flush is the number of 3 - card spade hands divided by the total number of 3 - card hands. So, $P=\frac{C(13,3)}{C(52,3)}=\frac{286}{22100}=\frac{11}{850}\approx0.0129$.

Answer:

a. 22100
b. 286
c. $\frac{11}{850}$