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# quiz7aANS - have an odd number of elements Remember that in...

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Math 240, Quiz 6 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1) How many strings of four decimal digits do not contain the same digit twice? There are 10 choices for the first place, 9 for the second, 8 for the third, and 7 for the fourth, so the total number is 10 · 9 · 8 · 7 = 10! 6! = P (10 , 4) = 5040 2) Suppose there are nine students in a math class. Show that the class must have at least five male students or at least five female students. Two ways to do this. A proof by contradiction would say: Suppose not. Then there are at most 4 females and 4 males in the class, for a total of 8 students, a contradiction. Thus, there must be at least 5 females or 5 males. To use the Pigeonhole Principle, say: There are 9 students, and 2 gen- ders. Mapping students to their genders, one gender must have ceilingleft 9 2 ceilingright = 5 people mapped to it, or there must be at least 5 of one gender. 3) Suppose the set A has 10 elements. How many subsets of A
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Unformatted text preview: have an odd number of elements? Remember that in subsets, order doesn’t matter. So the question becomes: How many ways can we choose an odd number of elements from A ? Well, there are ( 10 1 ) ways to choose 1 element, ( 10 3 ) ways to choose 3 elements, ( 10 5 ) ways to choose 5, ( 10 7 ) ways to choose 7, and ( 10 9 ) ways to choose 9. Add these together to get the correct answer. Another way to do it is to say: this is the sum of all the odd subsets of 10. By a corollary Prof. Miller proved in class, the number of odd subsets is equal to the number of even subsets. So the number of odd subsets is just the total number of subsets divided by 2. The total number of subsets of A has cardinality 2 1 0, so the number of odd subsets is 2 1 2 = 2 9 . Either way, you get 512 as your answer. Bonus (1 pt): Who was the leading scorer for the Wisconsin men’s basketball team during the NCAA tournament? Correct answer is Alando Tucker. 1...
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