# Thread: Ways to choose three people from a committe

1. ## Ways to choose three people from a committe

Given an eight-person committee, how many ways are there to choose the officer positions (chair person, vice-chairperson, and secretary) if a specific member of the committee (Angie) is either the vice-chairperson or she is not an officer?

There are a total of P(8, 3) = 8 * 7 * 6 = 336 ways to choose the officers without the last condition. However, given the condition the question has me stumped. Any suggestions?

2. ## Re: Ways to choose three people from a committe

Originally Posted by battery88
Given an eight-person committee, how many ways are there to choose the officer positions (chair person, vice-chairperson, and secretary) if a specific member of the committee (Angie) is either the vice-chairperson or she is not an officer?
Either she is or she is not.

$\displaystyle \mathcal{P}^7_2+\mathcal{P}^7_3$

3. ## Re: Ways to choose three people from a committe

Hello, battery88!

We must "talk" our way through this.

Given an eight-person committee, how many ways are there to choose the officer positions
(chair person, vice-chairperson, and secretary) if a specific member of the committee (Angie)
is either the vice-chairperson or she is not an officer?

It seems that Angie wants to be Vice-chair or nothing at all.
Very well, consider the two cases.

(1) Angie is Vice-chairman.
Then the other two offices (Chair, Secretary) can be filled in: $\displaystyle 7\cdot6 \,=\,42$ ways.

(2) Angie is not Vice-chairman. .(Angie is "out of the running".)
Then the three offices can be filled by the other seven members: $\displaystyle 7\cdot6\cdot5 \,=\,210$ ways.

Answer: .$\displaystyle 42 + 210 \:=\:210$ ways.

4. ## Re: Ways to choose three people from a committe

Thanks for the help. I can see it now!