FOM12 Permutations and Combinations Pretest
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Eve can choose from the following notebooks:
• lined pages come in red, green, blue, and purple
• graph paper comes in orange and black
If Eve needs one lined notebook and one with graph paper, which of the following pairs is not a
possible outcome?
A.
B.
C.
D.
____
2. A combination lock opens with the correct four-letter code. Each wheel rotates through the letters
A to L. How many different four-letter codes are possible?
A.
B.
C.
D.
____
20 736
48
1728
456 976
3. A combination lock opens with the correct four-letter code. Each wheel rotates through the letters
A to L. Suppose each letter can be used only once in a code. How many different codes are
possible when repetition is not allowed?
A.
B.
C.
D.
____
red and orange
black and blue
green and red
purple and black
20 736
11 880
1320
8976
4. A restaurant offers 60 flavours of wings and your choice of three dips. How many variations of
wings and dip can you order?
A.
B.
C.
D.
20
60
180
216 000
____
5. Evaluate.
A.
B.
C.
D.
____
6. Evaluate.
A.
B.
C.
D.
____
n
–n
n2
n3
8. Solve for n, where n  I.
A.
B.
C.
D.
____
13
16
20
23
7. Identify the expression that is equivalent to the following:
A.
B.
C.
D.
____
1 000 000
1 001 000
10 100 100
999 999
8
9
10
11
9. How many different permutations can be created when Anneliese, Becky, Carlo, Dan, and Esi line
up to buy movie tickets, if Esi always stands immediately behind Becky?
A.
B.
C.
D.
48
120
720
24
____
10. Evaluate.
14P7
A.
B.
C.
D.
____
11. How many numbers are there from 900 to 999 that do not have any repeated digits?
A.
B.
C.
D.
____
30 030
30 300
60 060
60 600
13. How many different arrangements can be made using all the letters in NUNAVUT?
A.
B.
C.
D.
____
81
90
100
72
12. Evaluate.
A.
B.
C.
D.
____
17 297 280
2 162 160
121 080 960
105 413 504
630
1260
2520
5040
14. How many different routes are there from A to B, if you only travel south or east?
A.
B.
C.
D.
10
20
40
8
____
15. Five quarters are flipped simultaneously. How many ways can three coins land heads and two
coins land tails?
A.
B.
C.
D.
____
16. There are 14 members of a student council. How many ways can 7 of the members be chosen to
serve on the dance committee?
A.
B.
C.
D.
____
21
15
30
42
18. Suppose that 10 teachers and 8 students volunteered to be on an environmental action committee.
The committee must consist of 2 teachers and 2 students. How many different environmental
action committees does the principal have to choose from?
A.
B.
C.
D.
____
1144
1716
3432
17 297 280
17. The numbers 10 to 16 are written on identical slips of paper and put in a hat. How many ways can
2 numbers be drawn simultaneously?
A.
B.
C.
D.
____
12
10
15
5
45
73
1260
5040
19. Which of the following is equivalent to
A.
B.
C.
D.
?
____
20. Identify the term that best describes the following situation:
Determine the number of arrangements of six friends waiting in line for movie tickets.
A.
B.
C.
D.
permutations
combinations
factorial
none of the above
Short Answer
1. A band sells shirts and CDs at their concerts. They have 5 CDs and there are 8 different styles of
shirt available in 5 sizes.
How many ways could someone buy a CD and a shirt?
2. The numbers 1 to 20 are written on slips of paper and put in a hat. How many possible ways can
you draw a either a prime number or a multiple of 6 from the hat?
3. Write the following expression using factorial notation.
4. Without calculating, predict which value is larger:
100P70 or 100P50
5. How many different routes are there from A to B, if you only travel south or east?
6. A fun fair requires 6 employees to help move one of the booths. There are 8 people available. How
many ways could a group of 6 be chosen?
Problem
1. Xtreme clothing company makes ski jackets in three colours (yellow, red, and silver) and sizes of
extra small, small, medium, large, and extra large.
a) Draw a tree diagram or an outcome table to determine how many different colour–size
variations of ski jackets the company makes.
b) Confirm your answer to part a) using the Fundamental Counting Principle.
2. Salima has 172 songs on her mp3 player.
a) How many different possibilities are there for the first four songs she hears, if she sets the player
to play the songs in random order without repeating? Show your work.
b) How does your answer change if repeating songs is allowed?
3. Explain why:
a) 29P0 =1
b) 29C0 =1
c) Why (a) and (b) are the same answer... aren’t permutations and combinations different?
FOM12 Permutations and Combinations Pretest
Answer Section
MULTIPLE CHOICE
1. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
3. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
4. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
5. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
6. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
7. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
8.
9.
10.
11.
12.
13.
14.
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
15. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
16. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
17. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
18. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
19. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
20. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
SHORT ANSWER
1. ANS:
200
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS:
11
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
3. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
4. ANS:
100P70
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
5. ANS:
56
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
6. ANS:
28
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
PROBLEM
1. ANS:
a)
There are 15 different colour–size variations.
Or a table:
Sizes
XS
S
M
L
XL
Colour
Yellow
Red
Yellow,XS Red,XS
Yellow,S
Red,S
Yellow,M Red,M
Yellow,L
Red,L
Yellow,XL Red,XL
Silver
Silver,XS
Silver,S
Silver,M
Silver,L
Silver,XL
b) The number of colour–size variations, C, is related to the number of colours and the number of
sizes:
C = (number of colours)  (number of sizes)
C=35
C = 15
There are 15 different colour–size variations.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS:
a) There are 172 songs and 4 positions they can be placed in.
Let A represent the number of arrangements:
There are 845 006 760 possible sets of four songs, without repetition.
b) With repetition, there are 172 possibilities for each position in the first four songs.
There are 875 213 056 possible sets of four songs, with repetition.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
3. ANS:
There is only one way to choose none of the items. And since none were chosen, order doesn’t
matter and it doesn’t matter if any elements were identical... they weren’t chosen!
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation