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sol10 - CS 330 Discrete Computational Structures Fall...

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CS 330 : Discrete Computational Structures Fall Semester, 2014 Assignment #10 Solutions Due Date: Sunday, Nov 23 Suggested Reading: Rosen Sections 6.1 - 6.3 These are the problems that you need to turn in. For more practice, you are encouraged to work on the other problems. Always explain your answers and show your reasoning. 1. [6 Pts] Elliott An ISU Computer Science shirt is sold in 9 colors, 5 sizes, striped or solid, and long sleeve or short sleeve. (a) How many diﬀerent shirts are being sold? (b) What if the black and yellow shirts only come in short-sleeve and solid? Solution: (a)By the product rule there are 9 × 5 × 2 × 2 = 180 diﬀerent types of shirt. (b)There are 7 × 5 × 2 × 2 = 140 diﬀerent types of shirts whose color is not black or yellow, and there are 2 × 5 × 1 × 1 = 10 diﬀerent types of shirts whose color is black or yellow. So the total shirts is 140 + 10 = 150. 2. [12 Pts] Elliott (a) How many diﬀerent four-letter codes can there be? (b) What if letters cannot be repeated? (c) What if letters cannot be repeated, and one of the initials is K? (d) What if letters can be repeated and at least one of the initials is K? Solution: (a) By the product rule there are 26 4 = 456976 diﬀerent four-letter codes. (b) If no letter is to be repeated, then there are 26 choices for the ﬁrst initial, 25 choices for the second, 24 choices for the third, and 23 choices for the fourth. By the product rule, the answer is 26 × 25 × 24 × 23 = 358800 . (c) We can subtract the number of four-letter code strings without K in it and also without repeating from the number of four-letter initial strings without repeating. Thus the answer is 26 × 25 × 24 × 23 - 25 × 24 × 23 × 22 = 55200 . (d) We can subtract the number of four-letter code strings without K in it and also with repeating from the number of four-letter initial strings with repeating. Thus the answer is 26 4 - 25 4 = 66351 3. [6 Pts] Peter How many integers between 10000 and 99999, inclusive, are divisible by 5 or 7? Solution: There are b 99999 5 c-b 10000 - 1 5 c = 19999 - 1999 = 18000 numbers between 10000 and 99999 that are divisible by 5. Similarly, there are b 99999 7 10000 7 c = 14285 - 1428 = 12857 multiples of 7. We need to remove the numbers that are counted twice, which is any number that is a multiple of 5 and 7, or in other words, a multiple of 35. There are b 99999 35 c-b 10000 35 c = 2857 - 285 = 2572 of these. Thus, there are 18000 + 12857 - 2572 = 28285 integers between 10000 and 99999 that are divisible by 5 or 7. 4. [8 Pts] Peter How many ways can 10 friends line up if Ann, Beth and Chris have to stand next to each other (a) where Ann is ahead of Beth and Beth is ahead of Chris? Solution:

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Since Ann, Beth, and Chris must be adjacent to each other and in a particular order, once we have placed any one of the three we know where the other two must go in line.
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• Fall '15
• Soma Chaudhuri
• ways, Polycephaly, Beth Chris

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