MATH 30-2
PERMUTATIONS, COMBINATIONS, &
THE FUNDAMENTAL COUNTING
PRINCIPLE - Module THREE
Module 3 - Assignment Booklet
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Date Submitted: ______________________
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Math 30-2:
Module Three Assignment
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Module Three Assignment
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Lesson 1: Fundamental Counting Principle
1. In Canada, all postal codes are made up of six characters, which follow the pattern letterdigit-letter-digit-letter-digit. The first letter of the code identifies a specific region. Each of
the remaining two letters in the postal code can be any of 20 letters, and each of the three
digits can be any digit from 0 to 9. All letters and digits can be repeated.
In Alberta, all postal codes start with the letter T. How many different postal codes are
possible in Alberta?
2. Each switch on a panel of 5 light switches may be in an “on” or “off” position at any given
time. One possible unique position of all 5 switches is shown.
Find the number of possible unique positions for all 5 light switches.
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3. In 1912, Alberta licence plates consisted of four digits. Each digit could be repeated, but
the first digit could not be zero. By 1941, Alberta licence plates consisted of five digits.
Each digit could be repeated, but the first digit could not be zero.
Compared with the number of licence plates available in 1912, find the increase in the
number of licence plates available in 1941.
4. Recall the example about Louie’s Bistro. You had a choice of 2 salads, mixed or Caesar.
You had a choice of 4 entrées: chili, club sandwich, roasted chicken, or grilled fish. Finally,
for dessert, you could choose a brownie, a cupcake, or a piece of cheesecake.
How many different meals are possible if you do not have to choose an item from each
category? Explain your solution.
5. Karen has a bank card that requires a 4-digit PIN number. All digits from 0 to 9 can be
used.
a. How many PINs are possible if the digits can repeat?
b. How many PINs are possible if the digits cannot repeat?
6. A group of 8 teenagers are lined up to buy tickets to a rock concert. How many possible
arrangements of teenagers are there? Leave your answer in factorial notation.
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Lesson 2: Factorial Notation
1. Simplify 9! .
6!
2. Solve the equation for n.
n ! = 42
(n - 2)!
3. Nine runners have made it to the zone cross-country running finals. Hariette is considered
the favourite to win based on her previous times. If she does win, in how many different
orders can the other racers finish?
4. A local hockey club is setting up an online system for registration for its players.
Seven-character passwords will be assigned to each player—each password will start with
the letter S to indicate the home club’s team name, Sharks. The other digits in the
password are made up of the numbers 4 through 9, and no digit can be repeated. How
many passwords can the hockey club create?
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Lesson 3: Problems Involving Permutations
1. Manny needs to create a 4-digit password for his cell phone. If no digits can be repeated, in
how many ways can this be done?
2. How many 3-letter arrangements can be made from the letters in VERTICAL if no letter can
be used more than once and each arrangement is made up of a vowel between two
consonants?
3. In a gymnastics club, 10 gymnasts practise forming a human pyramid. The pyramid uses
only 6 gymnasts at any one time. There are 3 gymnasts in the bottom row, 2 in the second
row, and 1 at the top.
Hemera/Thinkstock
If any gymnast can fill any position in the pyramid, how many different arrangements of
gymnasts are there?
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4. Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce,
1 poplar, 1 willow, and 1 elm. If the 4 birches are identical, then how many possible
arrangements of trees are there?
5. Solve for r.
5Pr =
60
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Lesson 4: Solving Problems That Involve Combinations
1. A car dealership is promoting a particular model of car that has 8 different optional features
available. Each optional feature can be purchased separately.
a. How many different packages of 3 optional features are possible for this model of the
car?
b. Explain why the number of different packages of 5 optional features is the same as the
number of different packages of 3 optional features for this model of car.
c. Another model of car has n different optional features available. When 2 optional
features are chosen for this model of car, there are 45 different packages available.
Determine algebraically the number of optional features, n, that are available for this
model of car.
2. a. How many diagonals are there in an octagon?
b. Why is 2.a. an example of a combination problem and not a permutation problem?
Math 30-2:
Module Three Assignment
3. How many 6-member committees can be formed from 8 girls and 9 boys if
a. there are no conditions
b. there must be exactly 3 girls
c. there must be at least 4 boys
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Lesson 5: Solving Probability Problems That Involve
Permutations and Combinations
1. A person has bought one ticket each for two separate lotteries. Determine the probability of
winning a grand prize for the two lotteries.
a. For the first lottery, 5 random numbers are drawn from 40. To win the grand prize, you
need to select the correct 5 numbers.
b. For the second lottery, 4 random numbers are drawn one at a time from 35. To win the
grand prize, you need to select the correct 4 numbers in the correct order.
c. A person is more likely to win the first lottery described in question 1.a., even though
the first lottery requires a person to have more correct numbers (5 compared to 4)
selected from a larger set of numbers (40 compared to 35). Explain why this is true.
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2. A zookeeper is setting up an African exhibit and has found the following animals available
from another zoo.
Antelope
Number
Available
Large Cats
Number
Available
Small Animals
Number
Available
common eland
1
African leopard
2
Abyssinian genet
4
greater kudu
1
Cape lion
1
common genet
2
nyala
2
Northwest African
cheetah
2
Cape ratel
6
red hartebeest
2
civet
1
black-backed jackal
2
side-striped jackal
3
a. If the zookeeper randomly selects an animal, what is the probability that a red
hartebeest will be chosen?
b. If the zookeeper randomly selects 8 animals, what is the probability that a cheetah or a
jackal will not be chosen?
c. If the zookeeper randomly selects 4 small animals, what is the probability that at least 1
Abyssinian genet will be chosen?
d. If the zookeeper asked for 4 antelope or large cats, what is the probability that both the
eland and the kudu would be received?
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3. a. Suppose you wrote the following message using the pigpen cipher from the beginning
of the lesson.
What is the probability that an interceptor would correctly guess the message on the
first try if the interceptor assumed each symbol corresponded to a letter of the
alphabet?
b. What is the probability of the person correctly guessing the following message on the
first try if that person knew
was an s?
(Hint: You can assume that if you guess the first
second
correctly.)
correctly, you will also guess the
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MODULE 3 SUMMARY
In this module you learned how to:
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Determine the number of ways in which events can occur.
You investigated the use of permutations and combinations in determining such
things as the possible match-up of teams for the playoffs, the number of codes
that can be used to open a garage, and the probability of winning a favourite
game.
How counting methods are used to solve problems involving codes and
passwords.
Some of the key ideas that you learned in each lesson are:
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how to apply the fundamental counting principle. If one task can be done in a
ways and a second in b ways, then the two tasks may be completed in a • b
ways.
how to apply and simplify factorial notation.
Factorial notation represents the product of consecutive natural numbers.
n! = n(n − 1)(n − 2)(n − 3)…(3)(2)(1)
how to use this knowledge to solve equations involving the use of permutation
and combination formulas.
how to apply factorial notation to problems such as determining the starting lineup for a baseball team.
that permutations are an arrangement of items when order is important.
you applied your knowledge of permutations to problems where order mattered,
such as a horse race.
that combinations are a selection of items where order is not important.
using combinations, you determined the number of chords that could be played
on a piano.
you investigated probability problems involving counting methods.
you analyzed problems to determine which counting method would be most
appropriate in determining the probability of an event happening.
In Module 4 you will explore how some problems can be modelled and solved using
polynomials
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MODULE 3 – PERMUTATIONS, COMBINATIONS, & THE
FUNDAMENTAL COUNTING PRINCIPLE SUMMATIVE ASSIGNMENT
Complete the following questions from your text book. Show steps completely and clearly, as
marks are assigned for mathematical literacy and communication. Always use graph paper,
rulers, and pencils as necessary. Attach questions and study notes securely to this booklet
before you hand everything in.
Text: Principles of Mathematics 12 - Chapter 2: Counting Methods
Section 2.1: Pages: 73 to 75 # 5, 8, 14
Section 2.2: Pages: 81 to 83 # 3all, 5e,f, 6a,b,e, 11a,c, 15
Section 2.3: Pages: 93 to 95 # 1d,f, 3a, 5, 8, 14, 15
Section 2.4: Pages: 104 to 107 # 6a, 9a, 11a, 13,
Section 2.5: Pages: 110 #2
Section 2.6: Pages: 118 to 120 # 4a,c,d, 8, 11a,b, 14aiii, 15a,c,18
Section 2.7: Pages: 126 to 128 # 3, 4, 5
Module 3 is now complete.
Once you have received your corrected work, review your instructor’s comments and prepare
for your module three test.