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Chapter 8: Problem 3
How many permutations of \(1,2,3,4,5,6,7\) are not derangements?
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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 them 16 times, every trio eight times, every foursome four times, each set of five twice, and each set of six once, but never all seven at once. If she had lunch every day during the 84 days of the conference, did she ever have lunch alone?
For which positive integers \(n\) is \(\phi(n)\) a power of \(2 ?\)
Professor Bailey has just completed writing the final examination for his course in advanced engineering mathematics. This examination has 12 questions, whose total value is to be 200 points. In how many ways can Professor Bailey assign the 200 points if (a) each question must count for at least 10 , but no more than 25 , points? (b) each question must count for at least 10 , but not more than 25 , points and the point value for each question is to be a multiple of 5 ?
Determine the number of integer solutions to \(x_{1}+x_{2}+x_{3}+x_{4}=19\) where \(-5 \leq x_{i} \leq 10\) for all \(1 \leq i \leq 4\).
Determine how many integer solutions there are to \(x_{1}+x_{2}+x_{3}+x_{4}=19\), if a) \(0 \leq x_{i}\) for all \(1 \leq i \leq 4\). b) \(0 \leq x_{1}<8\) for all \(1 \leq i \leq 4\). c) \(0 \leq x_{1} \leq 5,0 \leq x_{2} \leq 6,3 \leq x_{3} \leq 7,3 \leq x_{4} \leq 8\).
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