The Multiplication Principle of Counting Question: What is the multiplication principle of counting? To what type of situation is it applied?
Launch: Menu Entrees Chicken Beef Fish Tofu
Sides Desserts Rice Lemon Pie Baked Potato Cheesecake Steamed Vegetables
1) Look at the menu above. List out all possible orders you could make consisting of exactly one entree and one side. Do not add a dessert. A tree diagram will help.
2) You receive some good news. You can add a dessert to your order. List all possible orders you can make with one entree and one side and one dessert? Again, a tree diagram is useful. You can add desserts to the diagram you used in part 1.
In both questions above, the conjunction “and” was used. For example, you had a choice of one entree and one side and one dessert. How would the use of the conjunction “or” have changed the number of possible menu orders in question 2 above?
Investigation: Example 1: Choosing a vehicle Suppose you are ready to buy your first automobile. You can only buy one vehicle. The first step in the task of selecting your vehicle is to determine the model. As a class or small group, narrow your choices down to 4 models. List them in the table below.
As the second step of your vehicle selection process, you need to choose the exterior color. List 5 colors, you might choose for the exterior of your vehicle in the appropriate column of the table below. Vehicle Choices Model choices 1. 2. 3. 4.
Exterior color choices 1. 2. 3. 4. 5.
How many ways are there for you to complete the first step (choice of model)? How many ways are there for you to complete the second step (external color)? List all possible vehicles you could purchase by listing all possible model/external color combinations. You can choose to draw a tree diagram. How many different vehicle choices are there?
In the above setting, you were asked to choose both a model and an exterior color. What relationship do you notice between the number of total outcomes you counted and the number of choices you had for the two steps involved?
Would the number of possibilities have been different if you had first chosen the color and then chosen the model? Explain.
Based on what you noticed above, if there had been 7 model choices and 4 color choices, how many model/color combinations would you have had available to you?
Write an expression for the total number of model and color combinations you would have had if there had been m vehicle models and n exterior colors to choose from. Here m and n can represent any whole number. The relationship you have just discovered is known as the Multiplication Principle of Counting. This principle says that if a task involves two steps and the first step can be completed in m ways and the second step in n ways, then there are m*n ways to complete the task. This principle only applies in settings where the steps of the task are independent. For example, if some model choices had not had all color choices available then it would not have been possible to use this principle to count the vehicle choices. If the order of the steps is reversed in a two-step task where the steps are independent, does it change the number of ways to complete the task? What property of multiplication supports your answer?
Suppose you also have the chance to choose an interior color for our vehicle from Part 1. Recall you 4 models and 5 exterior colors to choose from. Additionally, you have 2 interior color choices. Add two interior color choices to the table below. Vehicle Choices Model choices 1. 2. 3. 4.
Exterior color choices Interior color choices 1. 2. 3. 4. 5.
Again, consider the vehicle choice process as a two-step task such as is described in the Multiplication Principle of Counting. The first step is to choose model and exterior color. How many ways did you determine this could be done? The second step is to choose interior color. How many ways are there to complete this second step? According to the Multiplication Principle of Counting, how many ways are there to accomplish the task of choosing a model and exterior color followed by the task of choosing an interior color?
You just found a way to consider a three-step task as a two-step task. In this way, you could apply the Multiplication Principle of Counting as it was presented. How did the total number of vehicle choices (model, exterior and interior color) relate to the number of choices for each of the threes steps in the task? Explain.
Write a version of the Multiplication Principle of Counting for a 3 step task by filling in the blank in the following statement. If a task involves three independent steps and the first step can be completed in m ways, the second step in n ways and the third step in p ways, then there are ________ ways to complete the task. Example 2: Pick a Number You are given the instructions to write down a 3 digit number. You can only use the digits 1 through 9. You are allowed to use a digit more than once. (Thus a number like 882 is ok). The task of selecting your 3 digit number can be broken down into three steps, namely the selection of each digit. Fill in the following table: Number of choices for first digit Number of choices for second digit after the first digit is chosen Number of choices for third digit after the first and second digits are chosen. Use the Multiplication Principle of Counting to determine the number of 3 digit numbers that can be made in this way. A restriction is added. You may not reuse a digit once it has been used. A number like 623 is still ok, but a number like 525 is not because the digit 5 is used twice. Should there be fewer or more possibilities now than in the previous problem? Explain. Fill in the table for this new problem. Number of choices for first digit Number of choices for second digit after the first digit is chosen Number of choices for third digit after the first and second digits are chosen. Use the Multiplication Principle of Counting to determine the number of 3 digit numbers that can be made with the new restriction on reusing digits in place.
Note: It is possible to extend the Multiplication Principle of Counting to count the possible outcomes for tasks with k steps where k is any natural number.
Example 3: Building a Sundae Students are allowed to build an ice cream sundae. Each student starts with a bowl of vanilla ice cream. Then each goes through a sundae bar where there is a choice of any of the following toppings. The students can build their sundaes in any way they choose. This includes the option of no toppings at all. Below is the beginning of a list of toppings on this bar. Fill in the 3 blanks with toppings you would include on such a bar. hot fudge, caramel, _______________, ______________, and ____________. For each student, the task of creating a sundae is a task involving five separate decisions. Below is the beginning of a tree diagram for building a sundae. The first two steps are filled in. Fill in the next branch of the diagram which is associated with step 3. Step 1 Step 2 Step 3
C aram el
N o c aram el
C aram el
N o Hot Fudge
N o c arm el
What sundae would you make? Can you see where that particular sundae choice would be represented on this tree diagram if it were completed? The tree diagram shows how the process of building a sundae can be thought of as a multi-step task. How many steps are there in the task for this problem setting? What are the steps and how many choices do you have for each?
Use the multiplication principle of counting to determine the number of possible sundaes the students can make at this sundae bar.
Think of another topping you would like to see on the list. What is it? How many sundaes could be made with this additional topping on the bar? What is the obvious advantage in applying the multiplication principle of counting in a setting where there are multiple steps and many choices for each?
Conclusions: Based upon what you have learned in this investigation, you should now be able to answer the following questions. In what type of situation is the multiplication principle of counting applied?
What does the multiplication principle of counting say about the number of ways to accomplish a two step task?
In general, how can you apply the principle to a multi-step task?
In Class Problems: It is very handy to use the Multiplication Principle of Counting in situations where tree diagrams or just listing the possibilities become too complicated and time-consuming. Use the Multiplication Principle of Counting to solve the following problems. 1. Suppose there are 23 people in attendance at a meeting. There are three door prizes to give away. The 23 people will put their names in a hat (one slip of paper per person). The names will be drawn out of the hat to award the prizes. a. This is a three step process. Each prize drawing will count as one step. Use
the Multiplication Principle of Counting to determine the number of ways the door prizes can be awarded if it is possible for the same person to be awarded more than one prize? In this setting, once a name is pulled from the hat, it is replaced before the next drawing. # of ways to award Prize 1 = _____ # of ways to award Prize 2 = _____ # of ways to award Prize 3 = _____ # of ways to award the three prizes = __________________. b.
The situation above is not typical. The typical situation is one in which a person’s name is removed from the hat and not replaced once he/she has won a prize. How many ways are there to award the three door prizes in this setting? # of ways to award Prize 1 = _____ # of ways to award Prize 2 = _____ # of ways to award Prize 3 = _____ # of ways to award the three prizes = __________________.
c. The situations in the parts a and b above are often categorized as selection “with replacement” and selection “without replacement”. Describe in your own words what the main difference is in a counting problem where the selection is done without replacement.
2. Suppose you run a business, and you need to issue passwords to your customers, so they can access an online system. Your passwords will have the form of two upper case letters (chosen from the 26 letter alphabet) followed by three digits (chosen from values 0 through 9). Here’s one example of a possible password: BZ022. Letter and digit choices can be repeated. a. How many steps are involved in the task of selecting a password?
b. List the steps and the number of choices for each.
c. Use the Multiplication Principle of Counting to determine how many passwords can be made using this format.
d. If your company has more than 1 million customers, will you have enough passwords using the format above? If not, suggest a change in the format of the passwords that would generate enough possibilities to cover 1 million or more customers? How many passwords can be generated using your new suggested format?
3. Six friends (Bailey, Deb, Katie, Nadia, Sara, and Tia) are about to occupy six seats in a movie theater row. The task of seating the group is a six-step task. a. If one wishes to count the number of ways the friends can occupy the seats. Would this be a problem counted “with replacement” or “without replacement”? Explain.
b. Use the Multiplication Principle of Counting to find out how many ways the six friends can occupy the six seats? Show your thought process.
c. If Katie absolutely insists on occupying seat #1, how many ways can the seating be accomplished? Show your thought process.
d. Katie is still determined to sit in seat #1. Tension develops and Bailey refuses to sit next to Katie. How many ways can the friends be seated given these requirements? Show your thought process.
Closure: What is the multiplication principle of counting? To what type of situation is it applied?
Homework: 1. a. Use the multiplication principle of counting to count the number of different three-letter “words” that can be constructed from our 26 letter alphabet. Suppose that letters may be repeated, so “dad” is a possible word. The combinations of 3 letters need not be actual words in any language. For example, “yxq” is just as much a possibility as “cat”.
b. Count the number of three letter “words” that can be constructed if it is not allowable to repeat a letter. (“dad” would not be an acceptable word in this scheme).
2. You get the chance to create your own pizza. You must first choose the size, (small, medium or large). Then you must choose the crust (thin or thick). Finally, you must choose only one topping. The topping choices are (pepperoni, sausage, anchovies, mushrooms, peppers, and black olives). a. How many different pizzas can you order? b. Great news!! You get two toppings. If you choose two different toppings then how many different pizzas can you order? c. Suppose the second topping could be the same as your first (for example you could order double anchovies). How many different pizzas are there now? 9
3. There are 16 members in a club. A slate of officers must be elected. There will be a president, vice president, secretary and treasurer. No person can hold two offices. a. How many ways can the offices be filled? b. Did you choose to do this problem with or without replacement? What words in the problem guided you in this decision?
4. A 5 member team consists of Bill, Ted, Juan, Shaq, and Kobe. It is necessary to choose 2 different members to go to a special camp. You need to find out how many possible pairs there are to send to camp. a. You decide to approach this problem as a two step process. First, you will choose one of the guys, and then you will choose a different guy. Apply the Multiplication Principle of Counting in this manner to see how many ways there are to choose one person followed by a different person to go to the camp.
b. List out all the possibilities counted in part a.
c. Looking at your list, you can see that the number of pairs to send to camp has been over counted. How many different pairs of players are there to send to camp?
d. What situation was present in this problem that caused the Multiplication Principle of Counting to over count the possible outcomes?
5. Read the following two problems. Do not try to solve them. Which problem would be appropriate for the Multiplication Principle of Counting? Which problem would be over counted by the Multiplication Principle of Counting? Explain your choice. Problem 1: A combination lock has whole numbers 0 through 39. A combination to unlock this lock consists of three numbers dialed in a specific order. Numbers may be reused. How many combinations are there?
Problem 2: A lottery drawing is performed by selecting one ball from each of three containers. There are ten balls in each container labeled with digits 0 through 9. The order in which the balls are does not matter. (This means 9, 1, 3 is the same as 3, 1, 9 or 1, 3, 9, etc). How many different three-ball drawings are possible?
6. Bethany goes to a hamburger restaurant. She is given her beef patty on a bun at the counter. She then goes to a bar to add toppings and condiments to her burger. Her choices at the bar are lettuce, tomatoes, pickles, onions, ketchup, mustard, mayonaise and jalapeno peppers. a. How many different burgers can she make at this bar?
b. If the restaurant is out of jalapeno peppers, how does this affect the number of choices she has available? 7. Challenge: You discover a language with a 4 letter alphabet. The letters are $, %, &, and @. Words in this language have 1 to 5 letters. There are no words with more than 5 letters. It is allowable to repeat letters within a word. The order of the letters in a word does matter. For example, the words “$%@” and “@%$” are different. How many different words can be made?