Question

www.g "A political discussion group consists of 6 Democrats and 10 Republicans. Three members are selected to attend a conference. Find the probability that the group will consist of all Republicans."

Answer 1

`Probability = (3\\)/(14)`

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 = (10!)/((10-3)!3!)`

`^(10)C_3 = (10!)/(7!3!)`

`^(10)C_3 = (10*9*8*7!)/(3!7!)`

Divide numerator and denominator by 7!

`^(10)C_3 = (10*9*8)/(3*2*1)`

`^(10)C_3 = (720)/(6)`

`^(10)C_3 = 120`

Next, the number of ways that 0 republicans can be selected from 6 will be calculated

`^6C_0 = (6!)/((6-0)!0!)`

`^6C_0 = (6!)/(6!0!)`

`^6C_0 = 1`

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 = (16!)/((16-3)!3!)`

`^(16)C_3 = (16!)/(13!3!)`

`^(16)C_3 = (16*15*14*13!)/(13!3!)`

`^(16)C_3 = (16*15*14)/(3!)`

`^(16)C_3 = (16*15*14)/(3*2*1)`

`^(16)C_3 = (3360)/(6)`

`^(16)C_3 = 560`

Lastly, the probability is calculated as follows;

`Probability = (^(10)C_3\ *\ ^6C_0)/(^(16)C_3)`

`Probability = (120\ *\ 1)/(560)`

`Probability = (120\\)/(560)`

Simplify fraction to lowest term

`Probability = (3\\)/(14)`