MM1D1 a
MM1P1 a,b
MM1P2 b
MM1P3 a,b
MM1P4 c
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.
3)
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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?
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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?
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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?
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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.
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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
Caramel
Hot Fudge
No caramel
Caramel
No Hot Fudge
No carmel
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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?
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Draft
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.
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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?
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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?
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Draft
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?
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?
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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?
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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?
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