Math 3201 Unit 2 Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. 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.
____
2. 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.
____
2
13
14
26
5. Evaluate.
A.
B.
C.
D.
____
360
380
420
480
4. 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.
____
3481
3540
3600
3660
3. The dinner special at a restaurant offers you a choice of 8 entrees, 2 salads, 5 drinks, and 3
desserts. How many different meals are possible if you choose one item from each category?
A.
B.
C.
D.
____
20 736
48
1728
456 976
1 000 000
1 001 000
10 100 100
999 999
6. Identify the expression that is equivalent to the following:
A.
B.
C. n3
D. (n + 1)!
____
7. Solve for n, where n
A.
B.
C.
D.
8
16
24
32
I.
____
8. Solve for n, where n
A.
B.
C.
D.
____
1440
4320
5040
2160
14. Evaluate.
A.
B.
C.
D.
____
1440
5040
360
720
13. 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.
____
72
100
81
90
12. How many ways can 7 friends stand in a row for a photograph if Sheng always stands beside his
girlfriend?
A.
B.
C.
D.
____
20
16
216
120
11. Suppose a word is any string of letters. How many two-letter words can you make from the letters
in LETHBRIDGE if you do not repeat any letters in the word?
A.
B.
C.
D.
____
17 297 280
2 162 160
121 080 960
105 413 504
10. 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.
____
13
15
17
18
9. Evaluate.
14P7
A.
B.
C.
D.
____
I.
330
660
990
1320
15. How many different arrangements can be made using all the letters in CALGARY, if the first letter
must be G?
A.
B.
C.
D.
360
480
120
720
____
16. How many different routes are there from A to B, if you only travel north or east?
A.
B.
C.
D.
____
17. 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.
____
130
126
122
118
21. Evaluate.
A.
B.
C.
D.
____
1000
715
635
808
20. Evaluate.
A.
B.
C.
D.
____
110
220
330
440
19. A fun fair requires 4 employees to work at the sack bar. There are 13 people available. How
many ways can a group of 4 be chosen?
A.
B.
C.
D.
____
1001
2002
6006
24 024
18. The numbers 1 to 11 are written on identical slips of paper and put in a hat. How many ways can
4 numbers be drawn simultaneously?
A.
B.
C.
D.
____
100
250
400
350
0
1
11
22
22. How many ways can 4 representatives be chosen from a hockey team of 17 players?
A.
B.
C.
D.
2380
57 120
31 060
9575
____
23. 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.
____
45
60
90
180
24. Which of the following is equivalent to
?
A.
B.
C.
D.
____
25. Solve for n.
nC1 = 30
A.
B.
C.
D.
____
26. 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
29. 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
28. 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.
____
permutations
combinations
factorial
none of the above
27. 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.
____
n=6
n = 10
n = 30
n = 60
164
254
336
1716
30. How many ways can the 6 starting positions on a hockey team (1 goalie, 2 defense, 3 forwards) be
filled from a team of 2 goalies, 5 defense, and 10 forwards?
A.
B.
C.
D.
1200
2400
4800
9600
Constructed Response 1.
1. 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.
2. 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?
3. 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?
4. 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?
5. Evaluate.
11 10 9!
6. Write the following expression using factorial notation.
8 7 6 5 4
7. Solve for n, and state the restrictions, where n
I.
8. How many different arrangements can be made using all the letters in VANCOUVER?
9. 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?
10. How many different routes are there from A to B, if you only travel south or east?
11. How many ways can you select 2 different flavours of ice-cream for a sundae if there are
16 flavours available?
Constructed Response 2.
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 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. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters
A to T.
a) Suppose each letter can be used only once in a code. How many different codes are possible
when repetition is not allowed?
b) How many more codes would there be if repetition is allowed?
3. Consider the word NUMBERS and all the ways you can arrange its letter using each letter only
once.
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.
4. 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.
5. 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.
6. Compare the number of different arrangements you can make using all the letters in the words
RED DEER and REGINA. Show your work.
7. From a group of seven students, four students need to be chosen for a graduation committee.
a) How many committees are possible? Show your work.
b) How many committees are possible, if only three students are needed on the committee?
c) Compare your answers for parts a) and b). What do you notice? Explain why this occurred.
8. A youth hostel has 3 rooms that contain 8, 5, and 3 beds, respectively. How many ways can the 16
players on a hockey team be assigned to these rooms? Show your work.
9. 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.
10. Three vehicles are taking a choir of 20 students to a recital. A minibus can take 12 students, an
SUV can take 5 students, and the remaining 3 students can ride with the choirmaster. How many
ways can the 20 students be assigned to the 3 vehicles? Show your work.
11. Solve each of the following and state the restrictions.
A. n-1P2 = 12
B. (n + 5)!
= 12
(n+3)(n+2)!
C. (n+2)! = 30
n!
Math 3201 Unit 2 Review
Answer Section
MULTIPLE CHOICE
1.
4.
7.
10.
13.
16.
19.
22.
25.
28.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
A
D
C
D
B
D
B
A
C
B
2.
5.
8.
11.
14.
17.
20.
23.
26.
29.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
C
B
C
A
B
A
B
A
C
C
3.
6.
9.
12.
15.
18.
21.
24.
27.
30.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
D
A
A
A
A
C
B
D
B
B
1. yes
2. 200
3. 3375
4. 360
5. 11! or 39 916 800
6.
7. 10
8. 181 440
9. 181 440
10.
SHORT ANSWER
11. 120
56
PROBLEM
1. a)
There are 15 different colour–size variations.
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.
2. a) There are 20 letters from A to T.
The number of different codes, C, is related to the number of letters from which to select on each
wheel of the lock, W:
C = W1 W2 W3
C = 20 19 18
C = 6840
There are 6840 different three-letter codes on this type of lock.
b) The number of different codes, R, is related to the number of letters from which to select on
each wheel of the lock, X:
R = X1 X2 X3
R = 20 20 20
R = 8000
R – C = 8000 – 6840
R – C = 1160
There are 1160 more codes if repetition is allowed.
3. 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.
4. There are 8 cars and 6 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 20 160 different ways.
5. 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.
6. RED DEER has 7 letters. There are 2 R’s, 3 E’s, and 2 D’s.
Let R represent the number of arrangements.
REGINA has 6 different letters.
6! = 720
More arrangements can be made using the letters in REGINA.
7. a) There are 7 students and 4 positions on the committee. Order does not matter.
There are 35 different committees possible.
b) There are 7 students and 3 positions on the committee. Order does not matter.
There are 35 different committees possible.
c) The answers to parts a) and b) are the same because n – r = r, so the two values in the
denominator are the same.
8. For the last room, there are 16 players and 3 beds. Order does not matter.
For the first room, there are now 13 players and 8 beds. Order does not matter.
The remaining 5 players share the middle room.
Using the Fundamental Counting Principle, the product of
the players can be assigned to the three rooms.
and
There are 720 720 ways to assign the 16 players to these rooms.
9. 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.
10. 12 of 20 students in the minibus:
5 of the remaining 8 students in the SUV:
3 students in the choirmaster’s car:
Let C represent the number of ways to assign rides to the 20 students:
There are 7 054 320 ways to assign rides to the 20 students.
11. a.Rest n≥2, n=13
b Rest: n≥-2, n=-8 reject, n=-1
c.Rest: n≥0,
n = -7 reject, n=4
is the number of ways