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Studying for a test? Prepare with these 8 lessons on Probability and combinatorics.
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A club of nine people wants to choose a board of three officers: a President, a Vice President, and a Secretary. How many ways are there to choose the board from the nine people? Now, we're going to assume that one person can't hold more than one office. That if I'm picked for President, then I'm no longer a valid person for Vice President or Secretary. So let's just think about the three different positions. So you have the President, you have the Vice President, VP, and then you have the Secretary. Now, let's say that we go for the President first. It actually doesn't matter. Let's say we were picking the President slot first and we haven't appointed any other slots yet. How many possibilities are there for President? Well, the club has nine people, so there's nine possibilities for President. There's going to be nine possibilities for President. Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices. Right? Because someone's going to be President, so one of the nine is going to be President. There's nine possibilities, but one of the nine is going to be President. So you take that person aside. He or she is now the President. How many people are left to be Vice President? Well, now there's only eight possible candidates for Vice President? Eight possibilities. Now he or she also goes aside. Now how many people are left for Secretary? Well, now there's only seven possibilities for Secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the 9 for President times the 8 for Vice President times the 7 for Secretary. You didn't have to do it this way. It could've been-- you could've picked Secretary first. Then there would've been nine choices. And then you could've picked Vice President, and there would've still been eight choices. And then you could've picked President last, and there would've only been seven choices. But either way you would've got 9 times 8 times 7. And that is, let's see, 9 times 8 is 72 times 7 is-- 2 times 7 is 14, 7 times 7 is 49 plus 1 is 50. So 504 possible ways to pick your board out of a club of only nine people.