hw-solutions-ch2 - MATH 3012 Homework Problems and...

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MATH 3012 Homework Problems and Solutions Chapter 2, Spring 2009, WTT and MTK Note: These problems are grouped into two categories. The first set consists of basic problems that everyone who expects to pass the course should be able to do. The second set consists of challenge problems. Students who would get a “C” would only be able to do a few of them, while students who would get an “A” would be able to do all—or nearly all—of them. Don’t be surprised if you are not able to do them all—as the course proceeds, we will occasionally slip in a real sleeper that no one on the planet knows how to solve! Group 1: Basic Problems - with solutions 1. How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if: a. Repetition of letters is allowed. Answer: 26 17 . This is a basic “license plate problem.” b. Repetition of letters is not allowed. Answer: P (26 , 17). Just counting permutations. c. Each word contains exactly 4 vowels (the letters a, e, i, o and u), with repetition of letters allowed. Answer: ( 17 4 ) 5 4 21 13 . Choose the four places where the vowels will be. The remaining 13 places will be consonants. Then the two sub-problems are license plates. d. Each word contains exactly 4 vowels, with repetition of letters not allowed. Answer: ( 17 4 ) P (5 , 4) P (21 , 13). Just like the preceding problem but with permutations. 2. A newsgroup requires members to login using a password that is a string of length 6, 7 or 8 characters with each character a letter of the English alphabet or a digit from 0 through 9. Also, the website distinguishes between upper and lower case letters. How many different passwords are possible? How may passwords are possible if it is required that at least one digit is used? Answer: Let n ∈ { 6 , 7 , 8 } . Then there are 62 n passwords of length n . Of these 52 n are “bad” because they contain no digits. So there are 62 n - 52 n “good” passwords of length n . In turn the total number of good passwords is: (62 6 - 52 6 ) + (62 7 - 52 7 ) + (62 8 - 52 8 ). 3. KK Bakery sells 16 varieties of donuts. Coffee can be ordered black, with milk, with sugar or with both. a. How may ways can a customer order a coffee and a donut? Answer: 4 · 16. Coffee choices are a bit string of length 2. For example, first bit is 0 for no milk and 1 for milk. Similarly, second bit is 0 for no sugar and 1 for sugar. Sixteen choices for the donut.
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This note was uploaded on 04/09/2009 for the course MATH 3012 taught by Professor Costello during the Spring '08 term at Georgia Tech.

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hw-solutions-ch2 - MATH 3012 Homework Problems and...

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