# 222a3 - them 16 times every trio eight times every foursome...

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MATH 222 FALL 2011, Assignment Three (due Thursday, Oct. 20 in class before the lecture begins) Show yourwork clearly. Illegible or disorganized solutions will receive no credit. 1. Find the number of integers between 1 and 10,000 inclusive which are (a) divisible by at least one of 3, 5, 7, 11; [3] (b) divisible by 3 and 5, but not by either 7 or 11. [3] 2. Professor J has just completed making the ±rst midterm test for his course in discrete mathematics. This test has 6 questions, whose total value is 30 points. In how many ways can professor J assign the 30 points if each question must count for at least 3, but not more than 6 points? [5] 3. At a 12-week conference in mathematics, Sharon met seven of her friends from college. During the conference she met each friend at lunch 35 times, every pair of
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Unformatted text preview: them 16 times, every trio eight times, every foursome four times, each set of ±ve twice, and each set of six once, but never all seven at once. If she has lunch every day during the 84 days of the conference, did she ever have lunch alone? [5] 4. How many derangements of 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 are there that (a) start with with 1 , 2 , 3 in some order? [2] (b) end with 4 , 5 , 6 , 7 in some order? [2] (c) start with with 1 , 2 , 3 in some order and end with 4 , 5 , 6 , 7 in some order? [2] 5. Let a n denote the number of n-digit numbers, each of whose digits is 1 , 2 , 3, or 4 and in which the number of 1’s is even. (a) Find a recurrence equation for a n . [3] (b) Find an explicit formula for a n . [3]...
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