Combinatorics problems

Transcription

1 Combinatorics problems Sections Math 245, Spring How to solve it There are four main strategies for solving counting problems that we will look at: Multiplication principle: A man s wardrobe consists of 5 sport coats, 3 dress slacks, and 2 pairs of shoes. Assuming they all match, in how many ways can he select an outfit? Answer: = 30. Addition principle: How many bytes have exactly 2 or exactly 3 zeros? Answer: = 84. Permutations: Six people are candidates for the positions of president and treasurer of a club. In how many different ways can the positions be filled? Answer: 6 5 = 30. (Note: order matters). Combinations: Six people are candidates for two committee positions (the two positions have the same duties). In how many different ways can the committee positions be filled? Answer: = 15. (Note: order does not matter). 2 Problems Now try these. If possible, specify which of the four strategies from above you are using to solve the problem. 1. A city council is composed of 5 liberals and 4 conservatives. A delegation of three is to be selected to attend a convention. How many delegations are possible? How many could have all liberals? How many could have 2 liberals and one conservative? 1

2 2. The school board consists of seven members, if the first person selected is the president, the second is the vice president and the third is the treasurer, how many ways can the officers of the board be chosen? 3. Two witnesses to a bank robbery had different memories of the license plate on the getaway car. Both agreed that the plate consisted of 6 digits. However, one noticed that there were exactly 2 ones, and the other noticed that there were exactly 3 nines on the plate. To be safe, and realizing that one of the witnesses might be mistaken, the police want to consider both possibilities. How many license plates consisting of 6 digits ( 0 to 9) have exactly 2 ones or 3 nines? 4. Three married couples (the Smiths, Joneses and Browns) sit on a bench. In how many ways can this be done if: a) there are no restrictions b) the Smiths sit at the ends c) the Smiths sit together d) each married couple sits together? 5. Three married couples (the Smiths, Joneses and Browns) sit at a circular table. In how many ways can this be done if: a) there are no restrictions b) the Smiths sit together c) each married couple sits together? 6. Each symbol in the Braille code is represented by a rectangular arrangement of six dots, each of which may be raised or at against a smooth background. Given that at least one of the six dots must be raised, how many symbols can be represented in Braille code? Is this a reasonable number (think about which symbols need to be represented)? 7. (a) For the usual 26 letter alphabet, how many strings of length 8 are there? (b) For the usual 26 letter alphabet, how many strings of length 8 are there, such that no letter is repeated? (c) For the usual 26 letter alphabet, how many strings of length 8 are there, such that the strings have the letter p in them? 8. In a certain state, license plates consist of from zero to three letters followed by from zero to four digits, with the provision, however, that a blank plate is not allowed. Suppose 85 letter combinations are not allowed because of their potential for giving offense. How many different license plates can the state produce? 2

3 9. A group of eight people are attending the movies together. (a) Two of the eight people insist on sitting side-by-side. many ways can the eight be seated together in a row? In how (b) Two of the people do not like each other and do not want to sit side-by-side. In how many ways can they be seated together in a row? This question is independent from the one in part (a). 10. An instructor gives an exam with twelve questions. Students are allowed to choose any ten to answer. (a) How many different choices of questions are there? (b) Suppose exam instructions specify that at most one of the questions 1 and 2 may be included among the ten. How many different choices of questions are there? (c) Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the ten or neither is to be included. How many different choices of ten questions are there? 11. In 2006, there were 250,844,644 registered vehicles in the US. If a uniform length of a licence place was required, and both letters and numbers were allowed in all positions, what should the number of symbols in a licence plate be to ensure that all vehicles get a unique licence plate, but that the number of options isn t too large? (Hint: guess and check would be a perfectly valid strategy in this problem.) 12. The below text has been taken from codes/ The format of an area code is NXY, where N is any digit 2 through 9 and X and Y are any digit 0 through 9. Initially, the middle digit of an area code had to be 0 or 1. When this restriction was removed in 1995, additional area code combinations became available. There are 800 possible combinations associated with the NXY format. Some of these combinations, however, are not available or have been reserved for special purposes. Among them are the following: Easily Recognizable Codes: When the second and third digits of an area code are the same, that code is called an easily recognizable code (ERC). ERCs designate special services; e.g., 888 for toll-free service. (For example, N11: These 8 ERCs, called service codes, are not used as area codes). 3

4 N9X The 80 codes in this format, called expansion codes, have been reserved for use during the period when the current 10-digit NANP number format undergoes expansion. 37X and 96X Two blocks of 10 codes each have been set aside by the INC for unanticipated purposes where it may be important to have a full range of 10 contiguous codes available. (a) How many area codes were available before 1995? Assume there were no other restrictions other than ones mentioned in the first paragraph. (b) Verify the figures in the text (800, 8, 80, and 10) using combinatorics. (c) How many area codes are in use? 3 Sample problems You do not have to solve these problems. They are examples of different types of problems: MULTIPLICATION PRINCIPLE How many even 3-digit positive integers can be written using the digits 1, 3, 4, 5, and 6? In how many ways can you select 4 cards, one after the other, from a 52-card deck: a) if the cards are returned to the deck after being selected? b) if the cards are not returned to the deck after being selected? How many 7-digit phone numbers can be created if the first digit cannot be 0 or 1, the second must be a 5, and the third must be a 3 or 4? How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0? PERMUTATIONS In how many ways can 11 books be arranged on a shelf: a) using all the books? b) using 4 of the books? In how many ways can the letters of the word MONDAY be arranged using a) all six letters? b) using 3 letters at a time? 4

5 COMBINATIONS A sample of 4 Ipods taken from a batch of 100 Ipods is to be inspected. How many different samples can be selected? In how many ways can a 5-card hand be dealt from a standard deck of cards? In how many ways can a committee of 7 be chosen from 9 girls and 8 boys if a) all are equally eligible? b) the committee must include 4 girls and 3 boys? 5