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Unformatted text preview: Discrete Structures Oct 17 2011 Assignment 7: Due at beginning of class Monday, Oct 17th Prof. Hopcroft *******Note: this homework is subject to point evaluations for clarity of writing and clarity of mathematical statements.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. (a) Use the principle of inclusion and exclusion to determine the number of integers between 1 and 1000, including 1 and 1000, not divisible by 2, 3, 5, or 7. Solution We begin by noting how many numbers are divisible by subsets of { 2 , 3 , 5 , 7 } . Because all numbers are primes, we can calculate how many numbers less than 1001 are divisible by all numbers in the subset, by taking the integer part of 1000 divided by the product of the numbers in the subset. divisible by all number divisible ∈ [1 , 1000] 2 500 3 333 5 200 7 142 { 2 , 3 } 166 { 2 , 5 } 100 { 2 , 7 } 71 { 3 , 5 } 66 { 3 , 7 } 47 { 5 , 7 } 28 { 2 , 3 , 5 } 33 { 3 , 5 , 7 } 9 { 2 , 5 , 7 } 14 { 2 , 3 , 7 } 23 { 2 , 3 , 5 , 7 } 4 Now by inclusion, exclusion, if there are an odd number of numbers in the subset we subtract how many integers are divisible by all elements of the subset. If there are an even number of numbers in the subset, we add how may integers are divisble by all elements. So we can add them up: number of integers ∈ [1 , 1000] not divisible by 2, 3, 5, or 7 = 1000 (500 + 333 + 200 + 142) + (166 + 100 + 71 + 66 + 47 + 28) (33 + 9 + 14 + 23) + 4 = 228 (b) If there are 100 students in a course where the possible grades are A, B, and C, what is the minimum number of students who must have gotten the same grade? What principle did you useminimum number of students who must have gotten the same grade?...
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