171sp07_supplement01sol - 550.171 Discrete Mathematics -...

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Unformatted text preview: 550.171 Discrete Mathematics - midterm #1 supplement SOLUTIONS/SKETCHSpring 2007 1. The following are all separate questions. (a) Ten boys are to be seated in a circular arrangement of chairs. The teacher can place the boys in any of the chairs, but she wants to make sure that Christopher and Gregory are not seated next to each other. How many different seating arrangements can the teacher make? Assume that two arrangements are the same if one is just the rotation of the other. Imagine the 10 seats are labelled as 1,2,3, ... ,9,10. Place Christopher in seat #1 (there is only one way to do this). There are now 9 empty seats labelled 2 through 8. Seat #2 and seat #8 are adjacent to seat #1, and Gregory cannot occupy these seats. Therefore, we can place Gregory in any one of the remaining 7 seats (i.e., seven choices to place Gregory). Once Christopher and Gregory have been placed, there are 8 seats remaining to place the rest of the 8 boys. There are 8! ways to do this. Thus, the total number of arrangements that keep Christopher and Gregory separated is 7...
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