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

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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

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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

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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

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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

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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

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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

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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

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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 10

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.

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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

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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:

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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.

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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.

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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

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