Name: ________________________ Class: ___________________ Date: __________ ID: A

2

Test 6: Perms and Combs REVIEW

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

____ 1. A multiple-choice test has 12 questions. Each question has 6 choices: A, B, C, D, E, or F. How many wayscan the test be answered?

A. 2 985 984 C. 72B. 18 D. 2 176 782 336

____ 2. The final score in a recreational soccer game is 7 - 4. How many scores are possible at the end of the firsthalf?

A. 28 C. 13B. 40 D. 18

____ 3. How many 4-digit numbers greater than or equal to 3000 and less than 8000 can be formed with no repetitionin their digits?

A. 5000 C. 120B. 5040 D. 2520

____ 4. At a school cafeteria, a meal consists of a main dish, a side dish, and a dessert. There are 3 main dishes, 4side dishes, and 7 desserts to choose from. How many different meals are possible?

A. 36 C. 45B. 84 D. 14

____ 5. In a geography class, 9 students are scheduled to give their presentations today. One student has to leaveearly for an appointment, so he will present first. How many different ways are there to schedule all oftoday’s presentations?

A. 362 880 C. 40 320B. 8 D. 3 628 800

Name: ________________________ ID: A

2

____ 6. A fast food restaurant offers hamburgers with a choice of 7 different condiments. A customer can choose anynumber of condiments. How many different combinations of condiments are possible?

A. 14 C. 128B. 49 D. 64

____ 7. A teacher has 8 different pencils in a mug on his desk. Each pencil can stand with the eraser-end up or withthe eraser-end down. How many different ways are there to stand the pencils in the mug?

A. 256 C. 64B. 16 D. 128

____ 8. How many 5-digit numbers can be formed that do not contain the digits 0 or 8?

A. 32 768 C. 40B. 99 968 D. 16 807

____ 9. A security code consists of 3 letters followed by 1 digit. The first letter in the code must be a vowel. Howmany different security codes are possible?

A. 33 800 C. 141 960B. 175 760 D. 3390

____ 10. A quarter is tossed 7 times. How many different sequences of outcomes are possible?

A. 128 C. 9B. 49 D. 14

____ 11. Which expression cannot be evaluated?

A. 8P 6 C. 9P 9

B. 10P 0 D. 12P 14

____ 12. What is the value of 10P 10 ?

A. 100 C. 0B. 3 628 800 D. 1

Name: ________________________ ID: A

3

____ 13. How many ways are there to arrange the letters in the word SHOE?

A. 10 C. 4B. 24 D. 6

____ 14. How many 2-letter permutations are there for the word LEARN?

A. 120 C. 6B. 20 D. 118

____ 15. In a youth hockey league, 9 teams compete for the championship. What is the number of ways that thewinner, second, third, and fourth place trophies could be awarded?

A. 3024 C. 504B. 15 120 D. 120

____ 16. What is the value of n for the equation nP 2 30?

A. n 210 C. n 6B. n 15 D. n 5

____ 17. How many permutations are there of the 5 digits in the number 85 697?

A. 120 C. 24B. 20 D. 5

____ 18. Which of these numbers has the least number of permutations of all its digits?

A. 445 869 C. 444 444B. 859 647 D. 444 484

____ 19. Which of these numbers has exactly 6 permutations of its digits?

A. 334 758 C. 333 333B. 748 536 D. 333 373

Name: ________________________ ID: A

4

____ 20. Which of these words has exactly 90 720 permutations of all its letters?

A. PERIDOTITE C. SERPENTINITEB. SANDSTONE D. GRANITE

____ 21. A coin is tossed 7 times. What is the number of ways the coin can land with 4 heads and 3 tails?

A. 5040 C. 144B. 2 D. 35

____ 22. There are 4 red plates and 2 green plates? How many different ways can the plates be stacked?

A. 15 C. 720B. 2 D. 48

____ 23. What is the value of 5C0 ?

A. 1 C. 0B. 2 D. 5

____ 24. How many combinations of 2 letters can be formed from the letters in the word SPRUCE?

A. 6 C. 12B. 15 D. 2

____ 25. How many 3-digit combinations are there of the digits in the number 39 517?

A. 5 C. 15B. 3 D. 10

____ 26. A student has 12 different books on her bookshelf. She wants to take 6 of them with her on a train trip. Howmany selections of 6 books could she make?

A. 665 280 C. 924B. 720 D. 72

Name: ________________________ ID: A

5

____ 27. In a children’s hockey league, 8 of the 20 teams advance to the playoffs. How many different groups of 8teams could end up in the playoffs?

A. 5 079 110 400 C. 125 970B. 160 D. 40 320

____ 28. There are 17 points on the circumference of a circle. How many lines can be drawn to connect all possiblepairs of points?

A. 289 C. 34B. 136 D. 272

____ 29. Which expression is not equivalent to 3C2 ?

A.3!

2! 3 2 !C. 3C1

B.3P 2

2!D. 2

3

____ 30. What is the value of the 6th number in row 11 of Pascal’s triangle?

A. 10C5 C. 11C6

B. 12C7 D. 5C10

____ 31. Which number in row n of Pascal’s triangle does the expression nC2 refer to?

A. 10th number C. 2nd numberB. 9th number D. 3rd number

____ 32. The number of different ways that 9 bikes can be locked in a bike rack isA. 3 628 800 C. 40 320B. 20 160 D. 362 880

Name: ________________________ ID: A

6

____ 33. An orchestra has 2 violinists, 3 cellists, and 4 harpists. Assume that the players of each instrument have to sittogether, but they can sit in any position in their own group. In how many ways can the conductor seat themembers of the orchestra in a line?A. 144 C. 24B. 72 D. 1728

____ 34. Solve for n in the expression nC4 nP4 117 600.

A. 8 C. 9B. 16 D. 8

____ 35. After the tryouts for the volleyball team, the coach selects 14 people to join the team. Due to a problem withtransportation, only 9 people can travel. In how many ways can the coach pick the people to go?A. 726 485 760 C. 630B. 2002 D. 126

____ 36. For a mock United Nations, 6 boys and 7 girls are to be chosen. If there are 12 boys and 9 girls to choosefrom, how many groups are possible?A. 846 720 C. 960B. 33 264 D. 120 708 403 200

____ 37. A scout troop is arranged in a circle for an opening ceremony with its leaders. Assuming there are 7 troopleaders who stand together for the ceremony and 39 scouts in total, including the troop leaders, in how manyunique ways can they be arranged around the circle?A. (32! )(7! ) C. (39! )(32! )B. 39C32 D. 39C7

____ 38. After a sports, tournament every player shakes hands with every other player once. If there are 36 handshakesin total, how many players are at the tournament?A. 18 C. 8B. 10 D. 9

Name: ________________________ ID: A

7

____ 39. Solve for n, where n I.n!

(n 1)! 4!

A. 8B. 16C. 24D. 32

____ 40. Evaluate.200P0

A. 0B. 1C. 2D. 200

Short Answer

1. A die has faces labelled 1 to 6. The number of outcomes when n dice are rolled is 1296. How many dicewere rolled?

2. How many numbers greater than 270 and less than 800 have three odd digits?

3. There are 9 routes from City A to City B and 6 routes from City B to City C. In how many ways is it possibleto travel from City A to City C via City B?

4. A fast food restaurant offers hotdogs with a choice of different condiments. A customer can choose anynumber of condiments. Suppose there are 512 possible combinations of condiments. How many differentcondiments does the customer have to choose from?

5. A pizza chain offers small, medium, and large pizzas, with a choice of thin or thick crust. There are 13different toppings available. How many different 1-topping pizzas are possible?

Name: ________________________ ID: A

8

6. A department store sells bicycles. There are 9 colours, 3 different seats, and the option of a basket. Howmany different bicycles are there to choose from?

7. How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and 8, if repetition is notallowed?

8. A password consists of 3 letters followed by 1 digit. Repetition is not allowed. How many passwords arepossible?

9. In how many ways can 7 magazines be arranged on a shelf?

10. Which of these words has the greater number of permutations of all its letters?BEAN or BEEN

11. How many permutations are there of all the letters in the word AUBURN?

12. A teacher has 4 spider plants, 4 cacti, and 5 geraniums. How many ways can he arrange the types of plants ina row on the window sill?

13. Solve this equation for n: nC3 84

14. A bakery has 15 different cakes and 6 different pies on display. You want to buy 3 cakes and 2 pies for aparty. How many selections could you make?

15. A manager is scheduling an employee for the next two weeks. The employee is to work 5 weekday shifts and2 Saturday or Sunday shifts. The employee has requested the second Friday off. How many ways can themanager arrange the employee’s schedule?

16. How many numbers are in row 11 of Pascal’s triangle?

Name: ________________________ ID: A

9

17. These are the numbers in row 8 of Pascal’s triangle. Use these numbers to generate the numbers in row 9.1 7 21 35 35 21 7 1

18. Show that 12C5 12C7 .

19. Simplify the expression(2n 2)!

(2n 2)!0!.

20. Solve for n: n(10P2) 10C8 .

21. Solve for r.34Pr = 34

22. How many different routes are there from A to B, if you only travel south or east?

23. How many different routes are there from A to B, if you only travel south or east?

Name: ________________________ ID: A

10

24. The numbers 1 to 16 are written on identical slips of paper and put in a hat. How many ways can 2numbers be drawn simultaneously?

25. How many 4-person committees can be formed from a group of 8 teachers and 5 students if theremust be either 1 or 2 teachers on the committee?

Problem

1. Use the digits 2, 3, 4, 5, and 6.a) How many 5-digit numbers can be formed when repetition is allowed?

b) How many 5-digit numbers can be formed when repetition is not allowed?

2. A city has a population of 18 176. Assume that each resident has a first name, a middle name, and a lastname. Explain why at least two residents must have the same first initial, the same middle initial, and thesame last initial.

3. Five couples go to see a movie. They sit together in 10 consecutive seats and couples sit together. Howmany seating arrangements are possible?

4. How many ways can the letters in the word PEANUT be arranged if the vowels always appear together?

5. A student has 13 books: 5 mystery, 6 science fiction, and 2 non-fiction. How many ways can the books bepositioned on a shelf if the books must stay with their genre?

6. The dance instructor must arrange 3 dances for the jazz club to perform at the dance recital. She has 6 dancesto choose from. How many different arrangements are possible?

7. There are 10 bottles of salad dressing on a shelf. Four of the bottles contain ranch dressing and the other 6bottles are all different. How many ways can the bottles be arranged in a row?

Name: ________________________ ID: A

11

8. A student must answer 4 of the 6 essay questions on an English exam.How many selections of questions are possible?

9. A chef wants to use leftover meats and vegetables to make a stew that contains 3 types of meat and 3 types ofvegetables. She has 6 types of meat and 7 types of vegetables to choose from. How many different stews canthe chef make?

10. a) What is the sum of all the numbers in row 4 of Pascal’s triangle?

What is the sum of all the numbers in row 5? In row 6?

b) What pattern do you see in your answers to part a?

How can you use this pattern to determine the sum of all the numbers in row n of Pascal’s triangle?

11. A standard deck of playing cards contains 4 suits (spades, clubs, diamonds, and hearts), each with 13 cards.a) How many different 5-card hands are possible?

b) How many different 5-card hands with only black cards are possible?

c) How many different 5-card hands are possible containing at least 3 black cards?

12. Arrange the following provincial capitals from the least to the greatest number of arrangements of letters:Fredericton, Charlottetown, Edmonton, Victoria, and Winnipeg.

Name: ________________________ ID: A

12

13. The chorus of a play has 17 females and 13 males. The director wishes to meet with 6 of them to discuss theupcoming production.a) How many selections are possible?

b) How many selections are possible if the group consists of three females and three males?

c) One of the male students is named Ajay. How many six-member selections consisting of Ajay, two othermales, and three females are possible?

14. Mr. Williams gives his class a test as part of their final mark for a course. Mr. Williams believes in choiceand offers some options to students. How many different options are possible ifa) students must complete three of the six questions in the first part and two of the three questions in thesecond part?

b) students must answer three of the questions in the first part and at least two of the three questions in thesecond part?

15. The locks on a briefcase open with the correct six-digit code. Each wheel rotates through the digits 0to 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.

16. How many different four-card hands that contain at least two face cards (jack, queen, or king) can bedealt to one person from a standard deck of playing cards? Show your work.

ID: A

1

Test 6: Perms and Combs REVIEWAnswer Section

MULTIPLE CHOICE

1. ANS: D PTS: 1 DIF: EasyREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

2. ANS: B PTS: 1 DIF: EasyREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

3. ANS: D PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

4. ANS: B PTS: 1 DIF: EasyREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

5. ANS: C PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

6. ANS: C PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

7. ANS: A PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

8. ANS: A PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

9. ANS: A PTS: 1 DIF: ModerateREF: 8.1 The Fundamental Counting PrincipleTOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

10. ANS: A PTS: 1 DIF: EasyREF: 8.1 The Fundamental Counting Principle LOC: 12.PCB1TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

ID: A

2

11. ANS: D PTS: 1 DIF: EasyREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial Theorem KEY: Conceptual Understanding

12. ANS: B PTS: 1 DIF: EasyREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial Theorem KEY: Procedural Knowledge

13. ANS: B PTS: 1 DIF: EasyREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

14. ANS: B PTS: 1 DIF: ModerateREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

15. ANS: A PTS: 1 DIF: ModerateREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

16. ANS: C PTS: 1 DIF: ModerateREF: 8.2 Permutations of Different Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial Theorem KEY: Procedural Knowledge

17. ANS: A PTS: 1 DIF: EasyREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

18. ANS: C PTS: 1 DIF: EasyREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial Theorem KEY: Conceptual Understanding

19. ANS: D PTS: 1 DIF: ModerateREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

20. ANS: B PTS: 1 DIF: ModerateREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

21. ANS: D PTS: 1 DIF: ModerateREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

22. ANS: A PTS: 1 DIF: ModerateREF: 8.3 Permutations Involving Identical Objects LOC: 12.PCB2TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

23. ANS: A PTS: 1 DIF: Easy REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge

ID: A

3

24. ANS: B PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

25. ANS: D PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

26. ANS: C PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

27. ANS: C PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

28. ANS: B PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

29. ANS: D PTS: 1 DIF: Easy REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding

30. ANS: A PTS: 1 DIF: Easy REF: 8.5 Pascal's TriangleLOC: 12.PCB4 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding

31. ANS: D PTS: 1 DIF: Easy REF: 8.5 Pascal's TriangleLOC: 12.PCB4 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding

32. ANS: D PTS: 1 DIF: Easy OBJ: Section 11.1NAT: PC1 TOP: Permutations KEY: fundamental counting principle

33. ANS: D PTS: 1 DIF: Average OBJ: Section 11.1NAT: PC2 TOP: Permutations KEY: fundamental counting principle

34. ANS: D PTS: 1 DIF: Average OBJ: Section 11.1 | Section 11.2NAT: PC2 | PC3 TOP: Permutations | Combinations KEY: permutations | combinations

35. ANS: B PTS: 1 DIF: Average OBJ: Section 11.2NAT: PC3 TOP: Combinations KEY: combinations

36. ANS: B PTS: 1 DIF: Difficult OBJ: Section 11.2NAT: PC3 TOP: Combinations KEY: combinations

37. ANS: A PTS: 1 DIF: Difficult + OBJ: Section 11.1NAT: PC1 TOP: PermutationsKEY: factorial | fundamental counting principle

38. ANS: D PTS: 1 DIF: Difficult + OBJ: Section 11.2NAT: PC3 TOP: Combinations KEY: combinations

39. ANS: C PTS: 1 DIF: Grade 12 REF: Lesson 4.2OBJ: 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 algebraicfraction containing factorials in both the numerator and denominator. | 5.4 Solve an equation that involvesfactorials. TOP: Introducing Permutations and Factorial NotationKEY: permutation | factorial notation

ID: A

4

40. ANS: B PTS: 1 DIF: Grade 12 REF: Lesson 4.3OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize strategiesfor determining the number of permutations of n elements taken r at a time.TOP: Permutations When All Objects Are Distinguishable KEY: permutation

SHORT ANSWER

1. ANS:4 dice were rolled.

PTS: 1 DIF: Moderate REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

2. ANS:75

PTS: 1 DIF: Moderate REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

3. ANS:There are 54 different ways.

PTS: 1 DIF: Easy REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

4. ANS:The customer has 9 different condiments to choose from.

PTS: 1 DIF: Moderate REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

5. ANS:78 different 1-topping pizzas are possible.

PTS: 1 DIF: Easy REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

6. ANS:There are 54 different bicycles to choose from.

PTS: 1 DIF: Easy REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding

ID: A

5

7. ANS:The number of 4-digit numbers that can be formed is 1680.

PTS: 1 DIF: Easy REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

8. ANS:There are 156 000 possible passwords.

PTS: 1 DIF: Moderate REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

9. ANS:There are 5040 possible arrangements.

PTS: 1 DIF: Easy REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

10. ANS:BEAN

PTS: 1 DIF: Easy REF: 8.3 Permutations Involving Identical ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding

11. ANS:360 permutations

PTS: 1 DIF: Moderate REF: 8.3 Permutations Involving Identical ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

12. ANS:90 090 ways

PTS: 1 DIF: Moderate REF: 8.3 Permutations Involving Identical ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

13. ANS:n 9

PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge

14. ANS:6825 selections

PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

ID: A

6

15. ANS:756 ways

PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge

16. ANS:11 numbers

PTS: 1 DIF: Easy REF: 8.5 Pascal's TriangleLOC: 12.PCB4 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding

17. ANS:The numbers in row 9 are:1 8 28 56 70 56 28 8 1

PTS: 1 DIF: Easy REF: 8.5 Pascal's TriangleLOC: 12.PCB4 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge

18. ANS:

L. S. 12!(12 5)!5!

12!7!5!

R. S. 12!(12 7)!7!

12!5!7!

12!7!5!

L.S. = R.S.

PTS: 1 DIF: Average OBJ: Section 11.2 NAT: PC2TOP: Combinations KEY: combinations

19. ANS:(2n 2)!

(2n 1)!0!

1 2 3 (2n 2)(2n 1)(2n)(2n 1)(2n 2)1 2 3 (2n 2)(2n 1) 1

(2n)(2n 1)(2n 2)

2n(4n 2 6n 2)

8n 3 12n 2 4n

PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1TOP: Permutations KEY: factorial

ID: A

7

20. ANS:n 10P2 10C8

n10!8!

10!2!8!

n 10!8!10!2!8!

n 12!

n 12

PTS: 1 DIF: Average OBJ: Section 11.1 | Section 11.2NAT: PC1 | PC2 | PC3 TOP: Permutations | CombinationsKEY: factorial | permutations | combinations

21. ANS:r = 1

PTS: 1 DIF: Grade 12 REF: Lesson 4.3OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize strategiesfor determining the number of permutations of n elements taken r at a time.TOP: Permutations When All Objects Are Distinguishable KEY: permutation

22. ANS:100

PTS: 1 DIF: Grade 12 REF: Lesson 4.4OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some elements arenot distinct. | 5.7 Explain, using examples, the effect on the total number of permutations of n elements whentwo or more elements are identical. TOP: Permutations When Objects Are IdenticalKEY: permutation | factorial notation

23. ANS:120

PTS: 1 DIF: Grade 12 REF: Lesson 4.4OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some elements arenot distinct. | 5.7 Explain, using examples, the effect on the total number of permutations of n elements whentwo or more elements are identical. TOP: Permutations When Objects Are IdenticalKEY: permutation | factorial notation

24. ANS:120

PTS: 1 DIF: Grade 12 REF: Lesson 4.5OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent the numberof arrangements of n elements taken n at a time, using factorial notation.TOP: Exploring Combinations KEY: counting | combination | factorial notation

ID: A

8

25. ANS:360

PTS: 1 DIF: Grade 12 REF: Lesson 4.6OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that involvepermutations 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 | Fundamental Counting Principle

PROBLEM

1. ANS:a)

Use the fundamental counting principle.There are 5 digits to choose from.There are 5 ways to choose each digit.So, the number of 5-digit numbers is: (5)(5)(5)(5)(5) 3125So, the number of 5-digit numbers that can be formed when repetition is allowed is 3125.

b)Use the fundamental counting principle.There are 5 ways to choose the first digit, 4 ways to choose the second digit,3 ways to choose the third digit, 2 ways to choose the fourth digit, and 1 way to choose the fifth digit.So, the number of 5-digit numbers is: (5)(4)(3)(2)(1) 120So, the number of 5-digit numbers that can be formed when repetition is not allowed is 120.

PTS: 1 DIF: Easy REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding | Communication | Problem-Solving Skills

2. ANS:Use the fundamental counting principle.For each resident, there are 26 choices for the first initial, 26 choices for the middle initial, and 26 choices forthe last initial.So, the number of possible choices for first, middle, and last initials is:(26)(26)(26) 17 576Since there are 18 176 residents and 18 176 > 17 576, at least 2 residents must have the same initials.

PTS: 1 DIF: Moderate REF: 8.1 The Fundamental Counting PrincipleTOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding | Communication | Problem-Solving Skills

ID: A

9

3. ANS:Use the fundamental counting principle.There are 10 consecutive seats.Anyone of the 10 people can sit in the first seat.Her or his partner must sit in the second seat.That leaves any one of 8 people to sit in the third seat.Her or his partner must sit in the fourth seat.That leaves any one of 6 people to sit in the fifth seat.Her or his partner must sit in the sixth seat.Continue the pattern for all 10 seats.So, the number of seating arrangements is: (10)(1)(8)(1)(6)(1)(4)(1)(2)(1) 3840There are 3840 possible seating arrangements.

PTS: 1 DIF: Difficult REF: 8.1 The Fundamental Counting PrincipleLOC: 12.PCB1 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication | Problem-Solving Skills

4. ANS:There are 3 vowels. The number of ways the vowels can be arranged is: 3!There are 3 consonants.Since the vowels must appear together, consider them as one object.So, there are 4 objects to arrange: the vowels and 3 different consonants.The number of permutations of 4 objects is: 4!So, using the fundamental counting principle:3! 4! 144

The letters can be arranged in 144 ways.

PTS: 1 DIF: Difficult REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication | Problem-Solving Skills

5. ANS:Determine the number of ways the books in each genre can be arranged on the shelf.Mystery books: 5! waysScience fiction books: 6! waysNon-fiction books: 2! waysThere are 3 genres. So, they can be arranged on the shelf in:3! ways3! 5! 6! 2! 1 036 800

There are 1 036 800 ways to position the books on the shelf.

PTS: 1 DIF: Difficult REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication | Problem-Solving Skills

ID: A

10

6. ANS:Use the permutation formula.

nP r n!

(n r)!Substitute: n 6 and r 3

6P 3 6!

(6 3)!

6!3!

120So, 120 arrangements of dances are possible.

PTS: 1 DIF: Easy REF: 8.2 Permutations of Different ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication

7. ANS:There are 10 bottles of dressing.Four of the bottles are identical.The number of ways the bottles can be arranged in a row is:10!4! 151 200

The bottles can be arranged in 151 200 ways.

PTS: 1 DIF: Easy REF: 8.3 Permutations Involving Identical ObjectsLOC: 12.PCB2 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding | Communication | Problem-Solving Skills

8. ANS:The order in which the questions are selected does not matter.There are 6 possible essay questions.The student must answer 4 of the questions.The number of ways of selecting the questions is:

6C4 15

There are 15 possible selections of questions.

PTS: 1 DIF: Easy REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication

ID: A

11

9. ANS:Number of ways of choosing meat:

6C3 6!

6 3 !3!

20Number of ways of choosing vegetables:

7C3 7!

7 3 !3!

35Use the fundamental counting principle:(20)(35) = 700So, the chef can make 700 different stews.

PTS: 1 DIF: Moderate REF: 8.4 CombinationsLOC: 12.PCB3 TOP: Permutations, Combinations and Binomial TheoremKEY: Conceptual Understanding | Procedural Knowledge | Communication | Problem-Solving Skills

10. ANS:a) Row 4:

1 + 3 + 3 + 1 = 8Row 5:1 + 4 + 6 + 4 + 1 = 16Row 6:1 + 5 + 10 + 10 + 5 + 1 = 32

b) The sum of the numbers in row 4 is 8, or 23 .

The sum of the numbers in row 5 is 16, or 24 .

The sum of the numbers in row 6 is 32, or 25 .

So, the sum of the numbers in row n of Pascal’s triangle is 2n 1 .

PTS: 1 DIF: Moderate REF: 8.5 Pascal's TriangleLOC: 12.PCB4 TOP: Permutations, Combinations and Binomial TheoremKEY: Procedural Knowledge | Conceptual Understanding | Communication | Problem-Solving Skills

ID: A

12

11. ANS:a) 52C5 2 598 960

b) There are 26 black cards (spades and clubs), so there are 26C5 or 65 780 possible hands with only black

cards.c) There are 3 possible situations:Case 1: all 5 cards are black:

26C5 65 780Case 2: 4 black cards, 1 red card

P(black) P(red) (26C4)(26C1)

388 700Case 3: 3 black cards, 2 red card

P(black) P(red) (26C3)(26C2)

845 000The total number of different 5-card hands containing at least 3 black cards is65 780 + 388 700 + 845 000 or 1 299 480.

PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3TOP: Combinations KEY: combinations

12. ANS:Winnipeg and Edmonton have the same number of combinations:

Winnipeg:8!

2!2! 10 080

Edmonton:8!

2!2! 10 080

Victoria:8!2! 20 160

Fredericton:11!2!2! 9 979 200

Charlottetown:13!3!2! 518 918 400

PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1TOP: Permutations KEY: factorial | permutations

ID: A

13

13. ANS:

a) 30C6 30!

(30 6)!6!

30!24!6!

593 775There are 593 775 possible ways of selecting 6 members of the chorus.

b) (17C3)(13C3) 17!(17 3)!3!

13!

(13 3)!3!

680(286)

194 480There are 194 480 possible ways of selecting three females and three males.c) There is only one way of selecting Ajay. This leaves 12 males left to choose the other two males.(1)(12C2)(17C3) 66(680)

44 880There are 44 880 possible ways to select the group that includes Ajay.

PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3TOP: Combinations KEY: combinations

14. ANS:a) The number of ways of selecting the questions in the first part is 6C3 or 20.

The number of ways of selecting the questions in the second part is 3C2 or 3.

The number of possible question selections is (20)(3) or 60.b) Since the students must answer at least 2 of the questions in the second part, they can answer two or threequestions.Case 1: Answer two questions in the second part(6C3)(3C2) 20(3)

60Case 2: Answer three questions in the second part(6C3)(3C3) 20(1)

20The number of ways of choosing three of the questions in the first part and at least two questions in thesecond part is 60 + 20 or 80.

PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3TOP: Combinations KEY: combinations | cases

ID: A

14

15. ANS:a) The number of different codes, C, is related to the number of digits from which to select on eachwheel of the lock, D:C = D1 D2 D3 D4 D5 D6

C = 10 10 10 10 10 10C = 1 000 000There 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 eachwheel of the lock, W:N = W1 W2 W3 W4 W5 W6

N = 10 9 8 7 6 5N = 151 200151200

1000000 100% 15.12%

Approximately 15% of these codes have no repeated digits.

PTS: 1 DIF: Grade 12 REF: Lesson 4.1OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize thefundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions made insolving a counting problem. | 4.4 Solve a contextual counting problem, using the fundamental countingprinciple, and explain the reasoning. TOP: Counting PrinciplesKEY: counting | Fundamental Counting Principle

ID: A

15

16. ANS:Case 1: exactly 2 face cards (and 2 not)There are 12 face cards and 40 other cards in a standard deck.

122

402

Case 2: exactly 3 face cards (and 1 not)123

401

Case 3: exactly 4 face cards124

Let H represent the number of hands with at least 2 face cards.

H 122

402

12

3

401

12

4

H 66 780 220 40 495H 60775

There are 60 775 different four-card hands that contain at least two face cards.

PTS: 1 DIF: Grade 12 REF: Lesson 4.7OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a contextualcounting problem, using the fundamental counting principle, and explain the reasoning. | 5.5 Determine thenumber of permutations of n elements taken r at a time. | 6.2 Determine the number of combinations of nelements taken r at a time. TOP: Solving Counting ProblemsKEY: counting | Fundamental Counting Principle | combination