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Our physics club has  members, among which we have  officers: President, Vice President, and Treasurer. However, one member, Alex, hates another member, Bob. How many ways can we fill the offices if Alex refuses to serve as an officer if Bob is also an officer? (No person is allowed to hold more than one office.)

 Jul 9, 2023
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First, we count the number of ways to fill the offices if Alex and Bob are both officers. There are 18 ways to choose the President, 17 ways to choose the Vice President, and 16 ways to choose the Treasurer, for a total of 18×17×16=4896.

Next, we count the number of ways to fill the offices if Alex is not an officer. There are 19 ways to choose the President, 18 ways to choose the Vice President, and 17 ways to choose the Treasurer, for a total of 19×18×17=5832.

Finally, we subtract the number of ways to fill the offices if both Alex and Bob are officers from the number of ways to fill the offices if Alex is not an officer, to get 5832−4896=936​.

 Jul 10, 2023

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