sol8 - U.C Berkeley — CS70 Discrete Mathematics and...

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Unformatted text preview: U.C. Berkeley — CS70: Discrete Mathematics and Probability Problem Set 8 Lecturers: Umesh Vazirani & Christos H. Papadimitriou Due October 20, 2006 at 4:00pm Problem Set 8 Solutions 1. Sampling The ways of sampling 2 from a 6-person family when order matters are 6! 4! = 6 · 5 = 30. If order does not matter, they are ( 6 2 ) = 15. 2. Win-loss Record In general if g is the number of games, the number of ways to end a season with w wins, t ties and l losses is g ! w ! t ! l ! Hence, the number of ways to end a season with w = 6, t = 2 and l = 1 is 9! 6!2!1! = 9 · 8 · 7 2 = 252 . The number of ways to end with w = 3, t = 3 and l = 3 is 9! 3!3!3! = 9! 6 3 = 1680 . 3. Numbers a) Consider 3-digit odd numbers with distinct digits first: you have 5 choices for the least significant digit, as the number has to be odd. Then, you have 8 choices for the most significant digit, as you cannot reuse the odd digit you already picked and 0 is not a valid choice at this position. Finally, you have 8 choices for the middle digit. Hence, there are 5you have 8 choices for the middle digit....
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sol8 - U.C Berkeley — CS70 Discrete Mathematics and...

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