MATH1061/7861 Assignment 10 Due 2pm Friday 30 May, 2008 Complete all of the following problems and hand in your solutions by the due date and time. Make sure that your name, student number and tutorial time are on each sheet of your answers. Solutions to all the problems will be distributed later. Late assignments will not be accepted unless you have a sucient documented reason, such as illness. 1. (a) What is the probability that a randomly chosen positive three-digit integer is a multiple of 6? (b) In a competition between players X and Y , the rst player to win three games in a row or a total of four games wins the competition. How many ways can the competition be played if X wins the rst game and Y wins the second and third? (Draw a tree.) (c) Six people attend the cinema and sit in a row with exactly six seats and an aisle at either end. (i) How many ways can they be seated together in the row? (ii) Suppose one of the six is a doctor who must sit on an aisle seat in case she is paged. How many ways can the people be seated together in the row with the doctor on an aisle seat? (iii) Suppose the six people consist of three married couples and each couple wants to sit together. How many ways can the six be seated together in the row so that couples are together? (d) Answer the following question from the textbook: Section 6.3, Q 27 (page 332-333). (Do not ask the lecturer to photocopy the textbook for you.) (e) Find the coecient of p4 q 6 in the expansion of (p q)10 . 2. Let S be the set of all bitstrings of length n, for some n 10. (a) Explain why | S |= 2n . (b) How many elements of S have both of their rst two bits equal to 0? (c) What is the probability that a randomly chosen element of S has its rst three bits the same as its last three bits? (d) We say that a bitstring is symmetric about its middle if the rst bit is the same as the last bit, the second bit is the same as the second last bit, and so on. If n is even, what is the probability that a randomly chosen element of S is symmetric about its middle? What if n is odd? (e) How many elements of S contain exactly zero, one or two bits equal to 0? (f ) What is the probability that a randomly chosen bitstring of length n has exactly two of its rst four bits equal to 0 and exactly three of its last six bits equal to 1? 3. Ten lazy MATH1061 students develop a scheme for doing their assignment. They decide that a team of four people can do the work, and the rest of the students can watch \"Dancing with the Stars\". In each part (a) to (g), the extra conditions apply only in that part. In answering each question, evaluate your answers in full, and show working. (a) How many dierent teams of size 4 can be selected from the 10 people? (b) Two of the students are keen on each other and are inseparable, so one will not be on the team without the other. Thus they must either both be on the team, or neither on the team. How many distinct selections of 4 people can be made? (continued over...) 1 (c) Lester is so attractive that girls cannot concentrate when they are with him. If four of the 10 are girls, and no girl can be on the team if Lester is on the team, then how many distinct selections of 4 people can be made? (d) If Darren and Bob are both on the team, then they will spend their whole time telling rude jokes, and the team will fail. If at most one of Darren and Bob can be on the team, then how many distinct selections of 4 people can be made? (e) Annette loves maths, and can't think of anything better to do than maths. What is the probability that she will be part of the team (assuming selection is random)? (f ) Two students love maths, ve students hate maths and three students don't care either way. How many distinct teams contain exactly one person who loves maths, two who hate maths and one who doesn't care either way? (g) Six of the students are male, and four are female. If selection is random, what is the probability that the team will contain at least one male, and at least one female? 4. (a) In Gold Lotto, you win Division 1 if your selection of 6 numbers out of 45 exactly matches the 6 numbers chosen by the lottery people (note that selection is without repetition). What is the probability that a single entry will win Division 1? (b) In a System n game, you choose n numbers out of 45, and win Division 1 if any 6 of those n numbers match those chosen by the lottery people. On a leading Australian lottery webpage it costs $0.70 to play a single game, and it costs $12994.80 to play a System 18 game. Explain how the System 18 entry fee would have been calculated. (c) In Powerball you select 5 numbers out of 45, then 1 number out of a dierent group of 45 (so repetition is allowed between the two groups). You win Division 1 if your selection of 5 and your selection of 1 exactly match equivalent selections by the lottery people. Find the probability that a single entry wins Division 1 in Powerball. (the end) 2