Math 3201
Chapter 2
Final Review
Multiple Choice
____
1. Eve can choose from the following notebooks:
• lined pages come in red, green, blue, and purple
• graph paper comes in orange and black
How many different colour variations can Eve choose if she needs one lined notebook and one with graph
paper?
A.
B.
C.
D.
____
2. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1 to 8.
How many different three-digit codes are possible?
A.
B.
C.
D.
____
20 736
11 880
1320
8976
6. A restaurant offers 60 flavours of wings. How many ways can two people order two different flavours?
A.
B.
C.
D.
____
20 736
48
1728
456 976
5. 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.
____
21
63
256
336
4. 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.
____
24
64
512
1024
3. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1 to 8.
Suppose each digit can be used only once in a code. How many different codes are possible when repetition
is not allowed?
A.
B.
C.
D.
____
6
8
12
16
3481
3540
3600
3660
7. A restaurant offers 60 flavours of wings. How many ways can two people order two servings of wings,
either the same flavour or different flavours?
A.
B.
C.
D.
3481
3540
3600
3660
June 2014
1
Math 3201
____
40 321
5041
40 123
16 777 217
12. Evaluate. (3!)2
A.
B.
C.
D.
____
2
13
14
26
11. Evaluate. 8! + 1!
A.
B.
C.
D.
____
432
576
646
720
10. How many possible ways can you draw a single card from a standard deck and get either a heart or a club?
A.
B.
C.
D.
____
20
60
180
216 000
9. The lunch special at a sandwich bar offers you a choice of 6 sandwiches, 4 salads, 6 drinks, and 3 desserts.
How many different meals are possible if you choose one item from each category?
A.
B.
C.
D.
____
Final Review
8. 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.
____
Chapter 2
8
9
18
36
13. Evaluate.
A. 0
B. 1
C. 3
D.
____
14. Evaluate.
A.
B.
C.
D.
____
1 000 000
1 001 000
10 100 100
999 999
15. Evaluate.
A.
B.
C.
D.
13
16
20
23
June 2014
2
Math 3201
____
Final Review
16. Identify the expression that is equivalent to the following:
A.
B.
C.
D.
____
Chapter 2
n
–n
n2
n3
17. Identify the expression that is equivalent to the following:
A.
B.
C. n2
D. n!
____
18. Identify the expression that is equivalent to the following:
A.
B.
C. n3
D. (n + 1)!
____
19. Solve for n, where n
A.
B.
C.
D.
____
____
I.
13
15
17
18
22. Solve for n, where n
A.
B.
C.
D.
I.
8
16
24
32
21. Solve for n, where n
A.
B.
C.
D.
____
10
19
20
39
20. Solve for n, where n
A.
B.
C.
D.
I.
I.
3
4
5
6
June 2014
3
Math 3201
____
23. Solve for n, where n
A.
B.
C.
D.
____
0
1
2
200
20
16
216
120
78 125
16 807
2520
1250
30. How many numbers are there from 1000 to 1999 that do not have any repeated digits?
A.
B.
C.
D.
____
441
420
399
2 097 152
29. Suppose a word is any string of letters. How many five-letter words can you make from the letters in
KELOWNA if you do not repeat any letters in the word?
A.
B.
C.
D.
____
48
120
720
24
28. Suppose a word is any string of letters. How many three-letter words can you make from the letters in
REGINA if you do not repeat any letters in the word?
A.
B.
C.
D.
____
49
128
720
5040
27. Evaluate. 200P0
A.
B.
C.
D.
____
8
9
10
11
26. Evaluate. 21P2
A.
B.
C.
D.
____
I.
25. 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.
____
Final Review
24. How many different permutations can be created when 7 people line up to buy movie tickets?
A.
B.
C.
D.
____
Chapter 2
504
1000
888
776
31. Solve for n. nP4 = 120
A.
B.
C.
D.
n=5
n=6
n=7
n=8
June 2014
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Math 3201
____
840
6720
13 440
1680
38. Evaluate.
A.
B.
C.
D.
____
48
72
140
180
37. Evaluate.
A.
B.
C.
D.
____
1440
4320
5040
2160
36. Evaluate.
A.
B.
C.
D.
____
1440
5040
360
720
35. How many ways can 8 friends stand in a row for a photograph if Molly, Krysta, and Simone always stand
together?
A.
B.
C.
D.
____
r=1
r=2
r=3
r=4
34. How many ways can 7 friends stand in a row for a photograph if Sheng always stands beside his girlfriend?
A.
B.
C.
D.
____
n=5
n=6
n=7
n=8
33. Solve for r. 9Pr = 72
A.
B.
C.
D.
____
Final Review
32. Solve for n. n – 2P2 = 30
A.
B.
C.
D.
____
Chapter 2
30 030
30 300
60 060
60 600
39. Evaluate.
A.
B.
C.
D.
330
660
990
1320
June 2014
5
Math 3201
____
0
1
4
16
46. Evaluate.
A.
B.
C.
D.
____
1001
2002
6006
24 024
45. Evaluate.
A.
B.
C.
D.
____
12
10
15
5
44. There are 14 members of a student council. How many ways can 4 of the members be chosen to serve on
the dance committee?
A.
B.
C.
D.
____
16
24
28
56
43. Five quarters are flipped simultaneously. How many ways can three coins land heads and two coins land
tails?
A.
B.
C.
D.
____
630
1260
2520
5040
42. How many different routes are there from A to B, if you only travel south or east?
A.
B.
C.
D.
____
120
180
360
720
41. How many different arrangements can be made using all the letters in NUNAVUT?
A.
B.
C.
D.
____
Final Review
40. How many different arrangements can be made using all the letters in CANADA?
A.
B.
C.
D.
____
Chapter 2
130
126
122
118
47. Evaluate.
A.
B.
C.
D.
0
1
11
22
June 2014
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Math 3201
____
1120
560
3360
1580
50. Suppose that 3 teachers and 6 students volunteered to be on a graduation committee. The committee must
consist of 1 teachers and 2 students. How many different graduation committees does the principal have to
choose from?
A.
B.
C.
D.
____
15
18
30
36
49. How many ways can 3 representatives be chosen from a soccer team of 16 players?
A.
B.
C.
D.
____
Final Review
48. Evaluate.
A.
B.
C.
D.
____
Chapter 2
45
60
90
180
51. Which of the following is equivalent to
?
A.
B.
C.
D.
____
52. Solve for n. nC1 = 30
A.
B.
C.
D.
____
53. 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
54. Identify the term that best describes the following situation:
Determine the number of codes for a lock with three dials numbered 0 to 9.
A.
B.
C.
D.
____
n=6
n = 10
n = 30
n = 60
permutations
combinations
factorial
none of the above
55. Identify the term that best describes the following situation:
Determine the number of pizzas with 4 different toppings from a list of 40 toppings.
A.
B.
C.
D.
permutations
combinations
factorial
none of the above
June 2014
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Math 3201
____
164
254
336
1716
59. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways can the
ten bikes be parked so that Elsa and Rashid are at either end of the bike rack?
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
58. How many ways can the 6 starting positions on a hockey team (1 goalie, 2 defense, 3 forwards) be filled
from a team of 1 goalie, 4 defense, and 8 forwards?
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
57. Identify the term that best describes the following situation:
Determine the number of two-card hands you can be dealt from a standard deck of 52 cards.
A.
B.
C.
D.
____
Final Review
56. Identify the term that best describes the following situation:
Determine the number of ways three horses can finish first, second, and third in a race of 12 horses.
A.
B.
C.
D.
____
Chapter 2
80 640
161 280
322 560
1 814 400
60. From a standard deck of 52 cards, how many different five-card hands are there with at least four black
cards?
A.
B.
C.
D.
388 700
649 740
1 299 480
454 480
Short Answer
1. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities when drawing a face card from a standard deck.
2. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities to choose from when buying a bicycle available in 4 sizes and 3
colours.
3. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities when picking a chair and a vice chair from a list of committee
members.
4. A band sells shirts and CDs at their concerts. They have 3 CDs and there are 4 different styles of shirt
available in small, medium, large, and extra large.
How many ways could you buy one CD and one shirt if you only consider one size of shirt?
5. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters A to O.
How many different three-letter codes are possible?
6. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters A to O.
Suppose each letter can be used only once in a code. How many different codes are possible when repetition
is not allowed?
June 2014
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Math 3201
Chapter 2
Final Review
7. The “Pita Patrol” offers these choices for each sandwich:
• white or whole wheat pitas
• 3 types of cheese
• 5 types of filling
• 12 different toppings
• 4 types of sauce
How many different pitas can be made with 1 cheese, 1 filling, 1 topping, and no sauce?
8. A theatre is showing 3 action movies, 4 comedies, 4 dramas, and 1 foreign film. How many choices does
Sophia have if she does not want to watch a drama or the foreign film?
9. A theatre is showing 2 action movies, 3 comedies, 3 dramas, 2 horror movies, and 2 foreign films. How
many choices does Sophia have if she does not want to watch an action movie or a horror movie?
10. Evaluate.
11. Evaluate. 11 10 9!
12. Evaluate.
13. Evaluate. 4! 3! 2!
14. Evaluate.
15. Write the following expression using factorial notation. 7 6 5 4 3 2
16. Write the following expression using factorial notation. 8 7 6 5 4
17. Write the following expression using factorial notation.
18. Write the following expression using factorial notation.
19. How many ways can you arrange the letters in the word FACTOR?
20. A baseball coach is determining the batting order for the nine players she is fielding. The coach has already
decided who will bat first and second. How many different batting orders are possible?
21. Solve for n, where n
I.
22. Solve for n, where n
I.
23. Solve for n, where n
I.
24. Solve for n, where n
I.
25. Evaluate. 5P3
26. There are nine different marbles in a bag. Suppose you reach in and draw one at a time, and do this three
times. How many ways can you draw the three marbles if you do not replace the marble each time?
27. Solve for n. nP2 = 90
28. Solve for r.
34Pr
= 34
29. Evaluate.
June 2014
9
Math 3201
Chapter 2
Final Review
30. How many different arrangements can be made using all the letters in VANCOUVER?
31. How many different arrangements can be made using all the letters in YELLOWKNIFE?
32. How many different arrangements can be made using all the letters in YELLOWKNIFE, if the first letter
must be L and the last letter must be Y?
33. How many different routes are there from A to B, if you only travel south or east?
34. A true-false test has ten questions. How many different permutations of answers can the teacher create if six
answers are true and four answers are false?
35. 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?
36. A fun fair requires 3 employees to sell tickets. There are 9 people available. How many ways could a group
of 3 be chosen?
37. Evaluate.
38. Evaluate.
39. How many ways can you select 2 different flavours of ice-cream for a sundae if there are 16 flavours
available?
40. How many 4-person committees can be formed from a group of 8 teachers and 5 students if there must be
either 1 or 2 teachers on the committee?
41. Solve for n. n + 4C1 = 14
42. Eight friends on bicycles find an empty bike rack with exactly eight spots for bikes. How many ways can
the eight bikes be parked so that Adya and Moira are parked at the ends of the rack?
43. A phys-ed teacher needs 4 equal teams for an activity. How many different ways can her class of 24
students be divided evenly into 4 teams?
44. How many ways can the top four cash prices be awarded in a lottery that sold 150 tickets if each ticket is
replaced when drawn?
45. How many ways can the top four cash prices be awarded in a lottery that sold 150 tickets if each ticket is
not replaced when drawn?
Problem
1. Hannah plays on a local hockey team. The hockey uniform has:
• four different sweaters: white, blue, grey, and black, and
• two different pants: blue and grey.
a) Draw a tree diagram to determine how many different variations of the uniform the coach can choose
from for each game are possible.
b) Confirm your answer to part a) using the Fundamental Counting Principle.
2. The locks on a briefcase open with the correct six-digit code. Each wheel rotates through the digits 0 to 9.
a) How many different six-digit codes are possible?
b) What percent of these codes have no repeated digits? Give your answer to the nearest percent.
3. Evaluate the following. Show your work.
4. Can you evaluate the following? Explain how you know. (–2)!
5. Consider the word NUMBERS and all the ways you can arrange its letter using each letter only once.
June 2014
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Math 3201
Chapter 2
Final Review
a) One possible permutation is ENBRUMS. Write three other possible permutations.
b) Use factorial notation to represent the total number permutations possible. Explain why your expression
makes sense.
6. At a used car lot, 8 different car models are to be parked close to the street for easy viewing, but there is
only space for 6 cars. How many ways can 6 of the 8 cars be parked in a row? Show your work.
7. At a used car lot, 5 different car models are to be parked close to the street for easy viewing. The lot has 4
red cars and 6 silver cars for the display. How many ways can the 5 cars be parked, if 2 red cars must be
parked at either end of a row of 3 silver cars? Show your work.
8. Mo has 12 new songs on his mp3 player. How many different 5-song playlists can be created from his new
songs, if no songs are repeated? Show your work.
9. A conveyor belt sushi restaurant lets you choose what to eat from different-coloured plates that go by. You
pay based on the colours of the plates.
After lunch, Ming has 2 red plates, 4 yellow plates, and 1 blue plate. How many ways can she stack her
plates in a single tower? Show your work.
10. A conveyor belt sushi restaurant lets you choose what to eat from different-coloured plates that go by. You
pay based on the colours of the plates.
After lunch, Ming stacks her 2 red plates and 4 green plates while Rudo stacks his 3 green plates, 1 red
plate, and 2 blue plates. Use the Fundamental Counting Principle to count how many ways they can make
the two stacks of plates. Show your work.
11. Two friends are building stacks of 12 coins. Stack 1 has 5 identical pennies, 3 identical nickels, and 4
identical quarters. Stack 2 has 3 identical pennies, 3 identical nickels, and 6 identical quarters.
Which set of coins can make more stacks of 12 coins? Show your work.
12. Two friends are building stacks of 15 coins. Stack 1 has 10 identical pennies, 3 identical nickels, and 2
identical quarters. Stack 2 has 5 identical pennies, 2 identical nickels, and 8 identical quarters.
Which set of coins can make more stacks of 12 coins? Show your work.
13. From a group of 12 students, three students need to be chosen for an environmental committee. How many
committees are possible? Show your work.
14. There are 12 boys and 15 girls in an English classroom. A group of 5 students is needed to read from a play.
If there are 2 roles for boys, 2 roles for girls, and a narrator who could be a boy or a girl, how many
different groups of 5 students are possible? Show your work.
15. There are 6 boys and 18 girls in a class. A group of 5 students is needed to work on a project. If at least 2
boys are needed, how many different groups of 5 students are possible? Show your work.
16. A hockey team is preparing for a group photo. The team has 2 goalie, 6 defense, and 8 forwards. The
photographer wants two rows of eight players. How many ways can the team arrange eight players in the
front row with at least one goalie? Show your work.
17. Fifteen camp counselors are signing up for training courses that have only a limited number of spaces. Only
5 people can take the water safety course, 4 people can take the first aid course, 3 people can take the
conflict management course, and 3 people can take the astronomy course. How many ways can the 15
counselors be placed in the four courses? Show your work.
June 2014
11
Math 3201
Chapter 2
Final Review
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
SHORT ANSWER
1. ANS:
2. ANS:
3. ANS:
4. ANS:
5. ANS:
6. ANS:
7. ANS:
8. ANS:
9. ANS:
10. ANS:
11. ANS:
12.
ANS:
13.
14.
15.
ANS:
ANS:
ANS:
16.
ANS:
17.
ANS:
18.
ANS:
19.
20.
21.
ANS:
ANS:
ANS:
B
C
D
A
B
B
C
C
A
D
A
D
C
B
B
C
B
A
B
C
C
A
C
D
D
B
B
D
C
A
A
D
B
A
B
D
D
A
B
A
B
C
B
A
C
B
B
A
B
A
C
C
C
D
B
A
B
C
A
D
no
yes
yes
12
3375
2730
360
7
8
54
11! or 39 916 800
288
4
7!
6! = 720
7! = 5040
10
June 2014
12
Math 3201
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
1.
ANS:
a)
Chapter 2
Final Review
13
4
3
60
504
n = 10
r=1
10
181 440
9 979 200
181 440
56
210
28
84
495
21
120
360
n = 10
1440
96 197 645 544
506 250 000
486 246 600
PROBLEM
There are 8 different variations of the hockey uniform to choose from.
b) The number of uniform variations, U, is related to the number of sweaters and the number of pants:
U = (number of sweaters) (number of pants)
U=4 2
U=8
There are 8 different variations of the hockey uniform to choose from.
2.
ANS:
a) The number of different codes, C, is related to the number of digits from which to select on each wheel of the lock, D:
C = D1 D2 D3 D4 D5 D6
C = 1 000 000
There are 1 000 000 different six-digit codes on this type of lock.
b) First determine the number of codes without repetition.
The number of different codes, N, is related to the number of digits from which to select on each wheel of the lock, W:
N = 10 9 8 7 6 5
N = 151 200
Approximately 15% of these codes have no repeated digits.
3.
ANS:
4.
5.
ANS:
ANS:
No. In the expression n!, the variable n is defined only for values that belong to the set of natural numbers.
a) Answers may vary. Sample answer: SBRUNME, NUMBRES, and BRUMENS
b) There are 7! possible permutations because there are 7 letters and 7 positions for them to occupy.
6.
ANS:
There are 8 cars and 6 positions they can be placed in. Let A represent the number of arrangements:
June 2014
13
Math 3201
Chapter 2
Final Review
The cars can be parked 20 160 different ways.
7.
ANS:
There are 4 red cars and 2 positions they can be placed in. There are 6 silver cars and 3 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 1440 different ways.
8.
ANS:
There are 12 songs and 5 positions they can be placed in. Let A represent the number of arrangements:
There are 95 040 different 5-song playlists that can be created from 12 songs
9.
ANS:
2 + 4 + 1 = 7 Let A represent the number of arrangements of 7 plates.
There are 105 different plate arrangements.
10.
ANS:
Let M represent the number of arrangements of Ming’s 6 plates.
Let R represent the number of arrangements of Rudo’s 6 plates.
Let A represent the number of arrangements of the two stacks.
There are 900 different plate arrangements.
11.
ANS:
Let A represent the number of arrangements of coins in stack 1.
Let B represent the number of arrangements of coins in stack 2.
More arrangements of coins can be made with stack 1.
June 2014
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Math 3201
12.
ANS:
Chapter 2
Final Review
Let A represent the number of arrangements of coins in stack 1.
Let B represent the number of arrangements of coins in stack 2.
More arrangements of coins can be made with stack 2.
13.
ANS:
There are 12 students and 3 positions on the committee. Order does not matter.
There are 220 different committees possible.
14.
ANS:
Let A represent the number of groups of 2 boys and 3 girls.
Let B represent the number of groups of 3 boys and 2 girls.
Number of groups of five = 30 030 + 23 100
Number of groups of five = 53 130
There are 53 130 groups of 5 students with those restrictions.
15.
ANS:
Let A represent the number of groups of 2 boys and 3 girls.
Let B represent the number of groups of 3 boys and 2 girls.
Let C represent the number of groups of 4 boys and 1 girls.
Let D represent the number of groups of 5 boys.
Number of groups of five = 12 240 + 3060 + 270 + 6
Number of groups of five = 15 576
There are 15 576 groups of 5 students with those restrictions.
16.
ANS:
Case 1: exactly 1 goalie
Case 2: exactly 2 goalies
Let H represent the number of arrangements of the eight players in the front row.
June 2014
15
Math 3201
Chapter 2
Final Review
There are 397 837 440 different ways to arrange eight players in the front row with at least 1 goalie.
17.
ANS:
5 of 15 people in water safety:
4 of the remaining 10 people in first aid:
3 of the remaining 6 people in conflict management:
3 people in astronomy:
Let C represent the number of ways to place the 15 counselors in the four courses:
There are 12 612 600 ways to place the counselors in the four courses.
June 2014
16