Ways to choose three people from a committe

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February 11th 2013, 06:33 AM
battery88

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?
February 11th 2013, 06:56 AM
Plato

Re: Ways to choose three people from a committe

Quote:

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.

$\mathcal{P}^7_2+\mathcal{P}^7_3$
February 11th 2013, 06:59 AM
Soroban

Re: Ways to choose three people from a committe

Hello, battery88!

We must "talk" our way through this.

Quote:

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: $7\cdot6 \,=\,42$ 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: $7\cdot6\cdot5 \,=\,210$ ways.

Answer: . $42 + 210 \:=\:210$ ways.
February 11th 2013, 07:03 AM
battery88

Re: Ways to choose three people from a committe

Thanks for the help. I can see it now!