Question

A council consists of five Democrats and seven Republicans. committee of six people is selected, find the probability of selecting throc and three Republicans

Answer 1

In order to find the probability you must start by finding the total possible outcomes for selecting a committee of six people from the 12 people.

`\begin{gathered} 12C6=(12!)/(6!(12-6)!) \\ 12C6=(12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)/(6\cdot5\cdot4\cdot3\cdot2\cdot1\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) \\ 12C6=(12\cdot11\cdot10\cdot9\cdot8\cdot7)/(6\cdot5\cdot4\cdot3\cdot2\cdot1) \\ 12C6=924 \end{gathered}`

then find the possible ways of finding selecting a committee of 3 republicans and 3 democrats

`\begin{gathered} 5C3\cdot7C3=(5!)/(3!(5-3)!)\cdot(7!)/(3!(7-3)!) \\ 5C3\cdot7C3=(5!)/(3!2!)\cdot(7!)/(3!4!) \\ 5C3\cdot7C3=10\cdot35 \\ 5C3\cdot7C3=350 \end{gathered}`

Then we divide the possible outcomes with the total outcomes to find the probability

`\begin{gathered} p=(350)/(924) \\ p=(25)/(66) \end{gathered}`

the probability is equal to 25/66