Answer:
[tex]Probability = \frac{3\\}{14}[/tex]
Step-by-step explanation:
Given
Republicans = 10
Democrats = 6
Total = Republicans + Democrats = 10 + 6 = 16
Selection = 3
Required
Probability that all selected members are Republicans
This implies that all selected members are republicans and none are republicans
This is calculated by (Number of ways of selecting 3 republicans * Number of ways of selecting 0 Democrats) / (Total number of possible selections)
First; the number of ways the 3 republicans from 10 can be selected needs to be calculated;
[tex]^{10}C_3 = \frac{10!}{(10-3)!3!}[/tex]
[tex]^{10}C_3 = \frac{10!}{7!3!}[/tex]
[tex]^{10}C_3 = \frac{10*9*8*7!}{3!7!}[/tex]
Divide numerator and denominator by 7!
[tex]^{10}C_3 = \frac{10*9*8}{3*2*1}[/tex]
[tex]^{10}C_3 = \frac{720}{6}[/tex]
[tex]^{10}C_3 = 120[/tex]
Next, the number of ways that 0 republicans can be selected from 6 will be calculated
[tex]^6C_0 = \frac{6!}{(6-0)!0!}[/tex]
[tex]^6C_0 = \frac{6!}{6!0!}[/tex]
[tex]^6C_0 = 1[/tex]
Next, the total number of possible selection will be calculated; In other words number of ways of selecting 3 politicians fro a group of 16
[tex]^{16}C_3 = \frac{16!}{(16-3)!3!}[/tex]
[tex]^{16}C_3 = \frac{16!}{13!3!}[/tex]
[tex]^{16}C_3 = \frac{16*15*14*13!}{13!3!}[/tex]
[tex]^{16}C_3 = \frac{16*15*14}{3!}[/tex]
[tex]^{16}C_3 = \frac{16*15*14}{3*2*1}[/tex]
[tex]^{16}C_3 = \frac{3360}{6}[/tex]
[tex]^{16}C_3 = 560[/tex]
Lastly, the probability is calculated as follows;
[tex]Probability = \frac{^{10}C_3\ *\ ^6C_0}{^{16}C_3}[/tex]
[tex]Probability = \frac{120\ *\ 1}{560}[/tex]
[tex]Probability = \frac{120\\}{560}[/tex]
Simplify fraction to lowest term
[tex]Probability = \frac{3\\}{14}[/tex]
