Question 1: How do you count choices using the multiplication principle?

Transcription

1 8.1 Permutations Question 1: How do you count choices using the multiplication principle? Question 2: What is factorial notation? Question 3: What is a permutation? In Chapter 7, we focused on using statistics to calculate probabilities. Using relative frequencies, we were able to calculate the likelihood of events using the history of what has happened in the past. However, in some applications we are interested in counting arrangements of objects in order to calculate the likelihood of a particular arrangement, For instance, suppose we wish to rank applicants for a job from a larger pool. How likely is it that the top five applicants are female? To answer this question, we need to be able to count the number of ways to rank five applicants from a larger pool. This will require us to learn about permutations. Permutations are used to calculate arrangements of objects where each objects are chosen without repetition. 1

2 Question 1: How do you count choices using the multiplication principle? A small cellular provider gives its customers 2 choices of phones to use. They may use an iphone or a phone that uses the Android operating system. In addition, the company offers three different calling plans: Budget plan, Regular plan, and the Deluxe plan. How many different choices of phone and calling plan does a customer have? To answer this question, we can use a decision tree and list out all of the choices a customer may make. Budget iphone Regular Deluxe Budget Android Regular Deluxe A decision tree show the different choice a customer makes when choosing a phone and plan. If we move left to right through the tree, we can list out each of the possibilities: iphone with Budget plan iphone with Regular plan iphone with Deluxe plan Android with Budget plan Android with Regular plan Android with Deluxe plan By listing out each of the possibilities, we see that there are six possible phone/plan choices. The decision tree helps us to list out these possibilities. However, if we only 2

3 need to know how many choices, we can multiply the number of choices for phones and plans Number of phones to choose from Number of plans to choose from This strategy is useful for determining the total number of choices even when there are a larger number of choices. Multiplication Principle Suppose we wish to know the number of ways to make n choices where there are d 1 ways to make choice 1 d 2 ways to make choice 2 d n ways to make choice n Then the total number of ways to make all of the choices is d d d 1 2 n Example 1 Multiplication Principle An online custom bicycle seller wishes to count the total number of different types of bicycles that are available through its website. The seller offers 4 different frame styles, 8 different fender colors, 10 3

4 different tire colors, 8 different wheel colors, 6 different pedal colors, and 12 different accessory colors. How many different bicycles can a customer order? Solution Each choice the customer must make leads to a different factor in the multiplication principle frame fender tire wheel pedal accessory style color color color color color 184,320 There are 184,320 different bicycles that can be ordered. Example 2 Multiplication Principle As the number of cars on the road has increased, so has the number of license plates. The format of the license plate determines how many different license plates there are. For each of the formats below, find the number of different license plates that are available. a. Three numbers Solution We use the multiplication principle and choose each number. There are ten choices for numbers 0 through 9 giving first second third number number number 1000 b. Three letters followed by three numbers Solution In this type of license plate, we have six choices to make. For each of the first three choices, there are 26 letters to choose from. For the last three choices, there are 10 numbers to choose from. This gives leads to the total number of license plates, 4

5 first second third first second third letter letter letter number number number 17,576,000 c. Six characters where each character may be a letter or number Solution Since the character can be a letter or a number, there are 36 choices for each character. This gives a total number of license plates, first second third fourth fifth sixth choice choice choice choice choice choice 2,176,782,336 5

6 Question 2: What is factorial notation? Certain patterns occur often when applying the multiplication principle. As we saw in Example 2, the factors that result from choices are often the same. In this case, we can use exponents to abbreviate the product: You may see the factors written with exponents instead of factors so it is important to recognize that they are the same. Another pattern that results from the multiplication principle can be written using factorial notation. Suppose a production line requires six workers to carry out six different jobs. Each worker can only do one job at a time. Once a worker is selected for a job, the other jobs must be carried out by the remaining workers. To find the number of ways we can assign workers to jobs, calculate the product first second third fourth fifth sixth job job job job job job 720 The number of ways to make each choice drops by one in each factor since each worker can only do one job. In effect, we can t choose the same worker twice. This is often indicated by saying that we want to assign workers without repetition. This type of product occurs so often that it is assigned its own symbol. 6

7 Factorial Notation For any positive integer n, n! n n1 n2 321 The value of 0! is defined to be 1. When we read an expression with factorial notation, a symbol like n! is read n factorial. Example 3 Use Factorial Notation Compute the value of each expression involving factorial notation. a. 6! Solution Use the formula above to get 6! b. 9! Solution It is tedious to multiply the factors out for larger numbers. Instead, use a calculator s factorial command to find the product. On a TI graphing calculator, start by typing 9. Then press ~~~ Í. Choosing 4 inserts the factorial symbol! from the PRB menu. The value is displayed on the screen. c. 100! 98! 7

8 Solution It is not practical to multiply all of the factors in the numerator and denominator. In addition, each of the factors in the fraction may not be calculated individually. If we try to do this the calculator will return an overflow error. Instead, write down some of the factor to see if any patterns emerge: 100! ! Every factor in the denominator is also in the numerator. These factors may be reduced to give 100! !

9 Question 3: What is a permutation? The term permutation refers to different arrangements in objects. We have already seen on example of a permutation in Question 2. When we allocated six workers among six jobs on a production line, we were counting the number of ways that we could arrange the workers among the jobs. This was a permutation of 6 workers taken 6 at a time. For permutations, we assume that the objects are arranged without repetition. In the context of the production line, this means that once a worker is given a job that worker cannot be assigned to another job. Let s modify this application slightly. How many ways can we assign six workers to four jobs without repetition? As before, we need to choose workers for each job first second third fourth job job job job 360 This is the same pattern that resulted in factorial notation, but we are missing the last two factors. However, we can still write this product in terms of factorial notation ! 6! ! 6 4! The number in the numerator indicates the number of objects we are selecting from. The number in the denominator is the difference between the number of object we are selecting from and the number we are selecting. This relationship leads to a general rule for permutations. 9

10 Permutations An arrangement of n objects taken r at a time without repetition is called a permutation. The number of these arrangements is symbolized Pn, r and found with P n, r n! n r! where r n. The symbol Pn, r is read the permutation of n objects taken r at a time. It can be confusing to use the letter P to indicate permutations and probability. For this reason, some textbooks will write P nr, or P instead of, n r P n r. In practice, this is less of a concern since the values in parentheses are numbers for permutations and events represented by capital letters in probability. In this text, we ll stick with writing permutations as Pn, r. If there is any possible confusion, we ll make sure to distinguish the permutations from the probabilities. Recall our initial production line example. In that example we wanted to find the number of ways to select six workers. In terms of permutations, this would be the permutation of six workers taken six at a time or P 6,6. This would be calculated 6! 6! 6! P6,6 6! 6 6! 0! 1 Remember that is defined to be equal to 1 For permutations, order is important. Rearranging the same workers among the jobs leads to a different arrangement since each job is different. If order did not make a difference, then there would only be one way to allocate the six workers to six jobs. This 10

11 might be the case if all of the workers were doing the exact same type of job. In this case, moving the workers to another job would give the same arrangement. To apply permutations, look for wording that indicates that order is important and that objects are selected without repetition. Example 4 Picking Stocks Suppose you wish to purchase two different stocks from Agilent Technologies Inc (A), Citigroup (C), and Ford (F). You will invest $10,000 in the first stock and $5000 in the second stock. a. How many ways are there to choose the two stocks? Solution Different amounts are being invested in each stock, This means that investing $10,000 in Citigroup and $5000 in Ford is different from investing $10,000 in Ford and $5000 in Citigroup. Therefore, order is important since it means that we invest different amounts. We also need to invest in two different stocks so we choose the stocks without repetition. We need to find the number of permutations of the three stocks taken two at a time, 3! P3, ! b. List all of the permutations of choosing 2 stocks from the list of three stocks. Solution To make the list easier to show, we ll write each possibility as a two-letter word using the stock symbols. The first letter in the word represents the stock we ll invest $10,000 in, The second letter represents the stock we ll invest $5000 in. The permutations are AC, AF, CA, CF, FA, FC 11

12 Example 5 Ranking Job Candidates A hiring committee is asked to rank the top five candidates for a teaching position from a pool of 20 candidates. How many different arrangements are there? Solution Since we want to rank the candidates, different arrangments must be counted differently. Additionally, once a candidate is ranked we cannot rank them in a different place. For these reasons, we need to calculate the permutation of 20 candidates taken 5 at a time, 20! 20! P20, ,860, ! 15! We can use a TI graphing calculator to find the same number by typing ÁÊ~~~Á Í. This pastes the permutation command npr from the MATH / PRB menu between 20 and 5. The calculator indicates that the number of objects must be typed first by placing the n in front of the P. Since the r is placed after P, the number that are taken at a time is typed after the permutation command npr. 12