Answer:
![Probability = \frac{3\\}{14}](https://tex.z-dn.net/?f=Probability%20%3D%20%5Cfrac%7B3%5C%5C%7D%7B14%7D)
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;
![^{10}C_3 = \frac{10!}{(10-3)!3!}](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20%5Cfrac%7B10%21%7D%7B%2810-3%29%213%21%7D)
![^{10}C_3 = \frac{10!}{7!3!}](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20%5Cfrac%7B10%21%7D%7B7%213%21%7D)
![^{10}C_3 = \frac{10*9*8*7!}{3!7!}](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20%5Cfrac%7B10%2A9%2A8%2A7%21%7D%7B3%217%21%7D)
Divide numerator and denominator by 7!
![^{10}C_3 = \frac{10*9*8}{3*2*1}](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20%5Cfrac%7B10%2A9%2A8%7D%7B3%2A2%2A1%7D)
![^{10}C_3 = \frac{720}{6}](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20%5Cfrac%7B720%7D%7B6%7D)
![^{10}C_3 = 120](https://tex.z-dn.net/?f=%5E%7B10%7DC_3%20%3D%20120)
Next, the number of ways that 0 republicans can be selected from 6 will be calculated
![^6C_0 = \frac{6!}{(6-0)!0!}](https://tex.z-dn.net/?f=%5E6C_0%20%3D%20%5Cfrac%7B6%21%7D%7B%286-0%29%210%21%7D)
![^6C_0 = \frac{6!}{6!0!}](https://tex.z-dn.net/?f=%5E6C_0%20%3D%20%5Cfrac%7B6%21%7D%7B6%210%21%7D)
![^6C_0 = 1](https://tex.z-dn.net/?f=%5E6C_0%20%3D%201)
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
![^{16}C_3 = \frac{16!}{(16-3)!3!}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B16%21%7D%7B%2816-3%29%213%21%7D)
![^{16}C_3 = \frac{16!}{13!3!}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B16%21%7D%7B13%213%21%7D)
![^{16}C_3 = \frac{16*15*14*13!}{13!3!}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B16%2A15%2A14%2A13%21%7D%7B13%213%21%7D)
![^{16}C_3 = \frac{16*15*14}{3!}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B16%2A15%2A14%7D%7B3%21%7D)
![^{16}C_3 = \frac{16*15*14}{3*2*1}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B16%2A15%2A14%7D%7B3%2A2%2A1%7D)
![^{16}C_3 = \frac{3360}{6}](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20%5Cfrac%7B3360%7D%7B6%7D)
![^{16}C_3 = 560](https://tex.z-dn.net/?f=%5E%7B16%7DC_3%20%3D%20560)
Lastly, the probability is calculated as follows;
![Probability = \frac{^{10}C_3\ *\ ^6C_0}{^{16}C_3}](https://tex.z-dn.net/?f=Probability%20%3D%20%5Cfrac%7B%5E%7B10%7DC_3%5C%20%2A%5C%20%5E6C_0%7D%7B%5E%7B16%7DC_3%7D)
![Probability = \frac{120\ *\ 1}{560}](https://tex.z-dn.net/?f=Probability%20%3D%20%5Cfrac%7B120%5C%20%2A%5C%201%7D%7B560%7D)
![Probability = \frac{120\\}{560}](https://tex.z-dn.net/?f=Probability%20%3D%20%5Cfrac%7B120%5C%5C%7D%7B560%7D)
Simplify fraction to lowest term
![Probability = \frac{3\\}{14}](https://tex.z-dn.net/?f=Probability%20%3D%20%5Cfrac%7B3%5C%5C%7D%7B14%7D)