Ÿ 8.1 The Multiplication Principle; Permutations The Multiplication Principle Example 1. Suppose the city council needs to hold a town hall meeting. The options for scheduling the meeting are either Monday, Tuesday, or Wednesday and either 6pm, 7pm, or 8pm. How many dierent meeting times are there? 1. We can list all of the possibilities: 2. Shortcut: A meeting time consists of two components (or choices): a day and a time. For the day, there are 3 options. For the time, there are also 3 options. Multiplication Principle 1. Suppose you have n components (at which you need to make a choice). Draw n slots, and label them underneath. 1 2... n 2. For each slot 1, 2,..., n, determine how many options there are. Write the appropriate number in each slot. m 1 m 2 1 2... m n n 3. Then, the total number of dierent ways to make the entire sequence of choices is the product of all of the n numbers. Total # of ways = m 1 m 2... m n 1
Exercise 2. Sarah is making beaded jewelry to sell. How many dierent styles can she make under the following conditions? 1. Suppose Sarah has 5 types of beads and wants to create a pattern that is 3 beads long. 2. Suppose Sarah has 5 types of beads and wants to create a pattern that is 3 beads long where the type of bead is not repeated in the pattern. Exercise 3. Six passengers are assigned to sit in row 9 (which consists of 6 seats) of an airplane on a ight from Pittsburgh to Philadelphia. How many dierent seating arrangements are there? Factorials The use of the multiplication principle often leads to products such as 6 5 4 3 2 1 (as in the previous exercise). If n is a natural number (i.e. a positive integer), then we use the notation n! for the product of all the natural numbers from n down to 1. We read n! as n factorial. Factorial Notation For any natural number n, n! = n (n 1) (n 2)... (3) (2) (1) Also, we have 1! = 1 0! = 1 Exercise 4. Louise owns a bakery and wants to arrange cupcakes in a line in her display case. If Louise wants to display 5 cupcakes, how many dierent arrangements are possible? 2
Exercise 5. Seven friends are competing in a race. April always comes in rst, and Greg always comes in last. How many dierent nishes are possible for these friends? Permutations Example 6. Consider the preceding exercise. What if a newspaper only wants to publish the results of the top three contestants? How does the problem change? This example can be generalized. Permutations are the dierent ways to arrange elements, without replacement or repetition, where the number of elements is greater than or equal to the number of slots. In Example 6, we had ve people for only two slots. We use the following notation to represent the total number of permutations: Permutations If P (n, r) (where r n ) is the number of permutations of n elements taken r at a time, then P (n, r) = n! (n r)! Remark 7. The letter P here stands for permutations. Do not confuse it with probability! You will always know when the P represents permutations because the parentheses will enclose two numbers separated by a comma. Exercise 8. 100 tickets are sold for a rae. Two tickets will be drawn (one at a time without replacement) to win two prizes. How many dierent combinations for winning tickets are there? 3
Exercise 9. A committee of four people is to be selected from a total of eight people. The committee will consists of a president, vice president, treasurer, and secretary. How many dierent arrangements are there for the committee? Exercise 10. Simplify the permutation formula for P (n, n). Is this the same result given by the multiplication principle? Distinguishable Permutations Example 11. Suppose Daniel has 3 apples and 2 pears to pack in his lunch for school one week (i.e. one piece of fruit for each day Monday thru Friday). How many permutations are there for bringing fruit to school? 1. First, what is the answer if each piece of fruit is dierent? (e.g. 1 apple, 1 orange, 1 banana, 1 pear, and 1 pomegranate) 2. What happens if we try to compute the answer to this problem using the same method (when each piece of fruit was dierent)? 3. Thus, we have to reduce our answer as follows: In the above example, we say that the pieces of fruit are not all distinguishable. We can generalize Example 11 as follows: 4
Distinguishable Permutations Suppose there are n dierent types of elements and a total of N elements. If there are m i elements in the i th class where m 1 + m 2 +... + m n = N, then the number of distinguishable permutations is N! m 1!m 2!... m n! This number is sometimes referred to as a multinomial coecient. In Example 11, n = 2 types of fruit, N = 5 pieces of fruit, m 1 = 3 apples, and m 2 = 2 pears. Exercise 12. Mike is a detective who managed to recover the license plate number of the car that belongs to a suspect in a murder investigation, except all of the letters/numbers are scrambled. Mike knows that the license plate number contains two H's, one 7, one J, and three A's. How many dierent license plate numbers are possible? Exercise 13. In how many ways can the letters in the following words be arranged? 1. Mississippi 2. conjugaters Exercise 14. Sarah is making beaded jewelry again. She has 4 red beads, 4 blue beads, and 1 white bead. How many dierent patterns can she create where all 9 beads are used? 5
Practice Problems Problem 15. A menu oers a choice of 3 salads, 8 main dishes, and 7 desserts. How many dierent meals consisting of one salad, one main dish, and one dessert are possible? Problem 16. Suppose Dan is a fan of J.R.R. Tolkien and C.S. Lewis. He has both the Chronicles of Narnia (7 books) and the Lord of the Rings Trilogy (3 books). If the only rule that Dan has for his bookcase is that all of the books by the same author stay together, how many dierent arrangements of the books are there? Problem 17. A baseball team has 19 players. How many 9-player batting orders are possible? Problem 18. A child has a set of toy blocks. There are 3 identical red blocks, 4 identical blue blocks, and 7 identical yellow blocks. If the child wants to line up the blocks in a row, how many distinguishable permutations are there? Problem 19. How many 7-digit telephone numbers are possible if the rst digit cannot be zero and 1. only odd digits may be used? 2. the telephone number must be a multiple of 10 (that is, it must end in zero)? 3. the telephone number must be a multiple of 100? 4. the rst 3 digits are 481? 5. no repetitions are allowed? 6