Introduction to Counting Homework Solutions

Transcription

1 Introduction to Counting Homework Solutions 1. Area codes: Here are the rules that were in use until 1993: The first digit must be 2 through 9. The second digit must be 0 or 1. The third digit must be 1 through 9. Three digit numbers with two zeros were special codes, ie. 700, 800 or 900. Three digit numbers ending with two ones were for special local codes, i.e. 411 for local directory assistance, 611 for repairs, etc. How many area codes were possible under those rules? Do not include the special codes or special local codes. Solution: There are 8 choices for the first digit, two for the second, and 9 for the third; = 144. Of these 8 are special codes 200, 300,..., 900 and 8 more are special local codes 211, 311,..., 911. Hence there were = 128 possible area codes. 2. Here are the options on a Raleigh Sports Comfort Bike, Model Year 2000 Sizes: 16.0, 18.0, 20.0, 22.0, L16, L19 The bikes come in men s and women s models. Both models come with the choice of two colors: Champagne Vapor, Mica Red Just to be prepared for walk-in customers, Bill s Bike Shoppe decides to have one of each possible model in stock. How many bikes must they order? Solution: 6 sizes, 2 genders, 2 colors: = (See the previous problem.) There are some extras available: Rack and Fender Mounts, two Water-Bottle Mounts (i.e. the bike comes with no water bottle mounts or else with two water bottle mounts), Clear Coat, CatEye Reflector Set. The reflectors can be easily added in the shop, but the others are best done at the factory. Again, assume that Bill wants to have one of every possible configuration. How many bikes must he stock? Solution: = Try to evaluate 80! on your calculator. What happens? Evaluate 80!/(78! 2!) without using a calculator. Solution: the number is too large for your calculator to represent. You get an error message. The quantity 80!/(78! 2!) reduces to 80 79/2 = How many code words of length three are possible? A code word of length three is any arrangement of three letters, repetitions allowed. For example, aab is a code word of length three. Solution: = How many code words of length three are possible using the letters a,b,c,d if we insist that the letter a must be followed by the letter b unless the letter a is the third letter? Solution: There are 43 such code words. Make a tree diagram.

2 7. A committee of 8 people wishes to choose a chairperson, a vice-chairperson, and a treasurer. How many lists of such officers are possible? Solution: There are 8 ways to choose the chairperson, then 7 ways to choose the vice-chairperson, then 6 ways to choose the treasurer = 336 ways to make a list. So there are 8. Charlie and Ed are going to play the best three out of five tennis games. That is, when one of them wins three games, then the series is over. How many outcomes are possible? Hint: Make a tree diagram. Solution: Make a tree diagram for Charlie, noting that the branch ends as soon as there are three wins or three losses. There are 20 possible outcomes. 9. A license plate has three letters followed by three digits. How many plates are possible? Solution: = Pat is going to buy a sound system. Pat can choose from 5 amplifier-tuners, 4 CD players, 7 speaker systems, and 3 turntables. Pat is going to include one amplifier-tuner, one CD player, and one speaker system. Pat has a vinyl collection but has not decided about including a turntable. How many different sound systems are possible? Solution: = There are four choices for the turntable. One of the choices is to not include a turntable. 11. A password must contain 6 or 7 letters followed by 2 digits. Repetitions of letters is allowed but the digits must be distinct. How many passwords are possible? Solution: The number of passwords with 6 letters and 2 digits is The number of passwords with 7 letters and 2 digits is Then take the sum of these two quantities There are 5 teams in the American League East, 5 teams in the American League Central, and 4 teams in the American League West. How many ways can a champion of each division plus a wild card team be chosen? Answer: The number of ways to choose playoff teams (order matters since the wildcard team is not a division champion) is After choosing the three champions is ways, there are 11 teams left to be the wild card team.

3 13. The WNBA has two divisions of eight teams each. Four teams from each division make the playoffs each year. Find the number of ways the WNBA playoff seeds can be filled (1-4 in each division). Answer: The four slots in each division can be filled in ways. Thus all together both divisions can be filled in ( ) 2 ways. 14. License plates in Ohio used to be three letters followed by three numbers (0-9). How many possible Ohio license plates are there? Answer: There are 26 ways to choose each number and 10 choices for each numerical digit, so all together there are = different license plates. 15. Recently Ohio moved to having three letters followed by four numbers. How many new license plates did this create? Answer: There are 10 times as many license plates now since we have four factors of 10. Thus the answer is In the basic Braille alphabet, each letter is represented by a pattern of raised dots as are 8 different punctuation marks, and the ten numerical digits are represented by the same patterns as the letters A-I. Each symbol is represented in a Braille cell which has a number of set positions for raised dots. How many different positions need there be to accommodate all 34 possible numbers and punctuation marks? Answer: In each possible place, there are two possibilities, either a raised dot or not. Thus we need a number of places so that We check that = 32 and = 64, so that there need to be 6 possible places. To find out more about Braille, you can visit the website

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