Chapter 4 and 5 Review



Chapter 4 and 5 Review

CH 4 and 5 Review

Short Answer

1. A Chinese restaurant features a lunch special with a choice of wonton soup or spring roll to start, sweet and sour chicken balls, pork, or beef for the main dish, and steamed or fried rice as a side dish. Create a tree diagram to show all the possible lunch specials at this restaurant. How many different possibilities are there?

2. On his university application, Enzo must list his course choices in order of preference. He must choose three of the four courses available in his major discipline, and two of the three courses offered in related subjects. In how many ways can Enzo list his course choices? Explain your reasoning.

3. Bruna lives in Hamilton and is planning a trip to Hong Kong. On the day she wants to travel, she can take one of two flights to Toronto, then one of three possible flights to Vancouver, and finally one of four flights available from Vancouver to Hong Kong. Use a tree diagram to determine the number of ways Bruna could fly from Hamilton to Hong Kong.

4. Kenya has a die and a coin. She tosses the coin and then rolls the die. Make a tree diagram to show the possible outcomes. How many different outcomes are there?

5. Bill works in an ice cream store for the summer months. How many different two- or three-scoop cones can Bill create if he has chocolate, mint chocolate chip, vanilla, maple walnut, and pistachio ice cream available?

6. Draw a tree diagram to illustrate the possible itineraries for Daima’s family if they can travel from their home in North Bay to Toronto by bus or train, and then from Toronto to Halifax by bus, train, or plane. How many possible itineraries are there?

7. In how many ways can the interviewers select a first, second, and third choice from a group of seven applicants for a position at a law firm?

8. How many ten-digit telephone numbers are available if none of the first four digits can be 0?

9. Evaluate the following.

a) [pic]

b) [pic]

c) [pic]

10. The track team has six students running in the 100-m dash. In how many ways can the students line up for the race?

11. Evaluate [pic] manually.

12. In how many ways can you arrange all the letters of the word POINTS?

13. In how many different orders can eight students present their culminating projects if John is first and Linda is last? Explain your reasoning.

14. Norma is creating a new game that has 15 different cards. In how many different ways can you deal out 5 cards from Norma’s deck?

15. How many six-digit even numbers less than 200 000 can be formed using all the digits 1, 1, 2, 2, 3, and 5?

16. In anticipation of a lunchtime rush, a small cafe has made seven egg-salad sandwiches, five tuna sandwiches, and six ham sandwiches, along with nine garden salads and six Caesar salads. In how many ways can this food be displayed in a single row on the counter?

17. A translation service has 33 employees, of whom 27 speak French, 11 speak Spanish, and 7 speak Hindi. Of the Spanish-speaking employees, 5 speak Hindi and 6 speak French, but only 3 of the French-speaking employees speak Hindi. All of the employees speak at least one of these three languages. Use a Venn diagram to determine how many of the employees speak all three.

18. The Statsville hockey team has nine forwards, nine defenders, and two goalies. Use combinations to determine the number of ways the team can form a line-up of three forwards, two defenders, and one goalie.

19. How many different sums of money can you make with four pennies, two nickels, and six quarters?

20. Use Pascal’s triangle to expand (x + y)6.

Problem

You are thinking of using some leftover ceramic tiles to make a decorative border for a patio. You have ten blue, four gold, and six white tiles.

21. In how many ways can you make the border if you need nine tiles and use equal numbers of all three colours? Explain your reasoning.

22. Of the 190 students graduating from Statsville High School, 111 played on a sports team, 26 served on the students’ council, and 67 were members of clubs. Only 10 students did not participate in any of these activities. If 13 students were members of both a team and a club, 9 were on both a team and the students’ council, and 11 were members of both the students’ council and a club, how many graduating students have been a member of a team as well as the students’ council and a club? Illustrate your answer with a Venn diagram.

23. An athletic committee with three members is to be selected from a group of six gymnasts, five weightlifters, and seven sprinters. In how many ways can the committee be selected if it must include at least one weightlifter? Explain your reasoning.

24. The students producing a school fashion show plan to have five scenes with music between them. The music students have come up with 18 pieces: 6 for piano, 5 for recorder, and 7 for guitar. The students want to use at least 1 piece for the piano. In how many ways can the group choose the 4 pieces of bridging music? Explain your reasoning.

25. In the expansion of (ax + by)8, the first three terms are 256x8, 6144x7y, and 64 512x6y2. Find the values of a and b.

CH 4 and 5 Review

Answer Section

SHORT ANSWER

1. ANS:

[pic]

There are 12 possible meals at this Chinese restaurant.

REF: Applications OBJ: Section 4.1 LOC: OD2.02 TOP: Tree diagrams

2. ANS:

Enzo has 144 ways to choose his courses since he can choose his major courses in 4 × 3 × 2 ways and his options in 3 × 2 ways. By the multiplicative rule, he has 24 × 6 = 144 ways in which he can rank the courses.

REF: Applications OBJ: Section 4.1 LOC: CP1.02 TOP: Fundamental counting principle

3. ANS:

[pic]

There are 24 ways Bruna could choose her flights.

REF: Applications OBJ: Section 4.1 LOC: OD2.02 TOP: Tree diagrams

4. ANS:

[pic]

There are 12 possible outcomes.

REF: Applications OBJ: Section 4.1 LOC: OD2.02 TOP: Tree diagrams

5. ANS:

There are 80 ways in which Bill can create two- or three-scoop cones.

REF: Applications OBJ: Section 4.1 LOC: CP1.02 TOP: Fundamental counting principle

6. ANS:

[pic]

There are six possible itineraries for Daima and her family.

REF: Applications OBJ: Section 4.1 LOC: OD2.02, CP1.02

TOP: Tree diagrams

7. ANS:

There are 210 ways to select the top three choices.

REF: Applications OBJ: Section 4.1 LOC: CP1.02 TOP: Fundamental counting principle

8. ANS:

There are 6 561 000 000 telephone numbers possible.

REF: Applications OBJ: Section 4.1 LOC: CP1.02 TOP: Fundamental counting principle

9. ANS:

a) 604 800

b) 7.602 692 257 × 1014

c) 1.157 655 162 × 1016

REF: Knowledge & Understanding OBJ: Section 4.2 LOC: CP1.04

TOP: Factorials

10. ANS:

The students can line up in 6!, or 720 ways.

REF: Applications OBJ: Section 4.2 LOC: CP1.02, CP1.03

TOP: Permutations

11. ANS:

2730

REF: Knowledge & Understanding OBJ: Section 4.2 LOC: CP1.04

TOP: Factorials

12. ANS:

You can arrange the letters in 6! = 720 ways.

REF: Applications OBJ: Section 4.2 LOC: CP1.02, CP1.03

TOP: Permutations

13. ANS:

There are six students making presentations between John’s and Linda’s. These six students can appear in 6P6 = 720 different orders.

REF: Applications, Communication OBJ: Section 4.2 LOC: CP1.02, CP1.08

TOP: Permutations

14. ANS:

3 603 600

REF: Applications OBJ: Section 4.2 LOC: CP1.02 TOP: Permutations

15. ANS:

24

REF: Thinking/Inquiry/Problem Solving OBJ: Section 4.3 LOC: CP1.05

TOP: Permutations with some identical items

16. ANS:

[pic]

REF: Applications OBJ: Section 4.3 LOC: CP1.05

TOP: Permutations with some identical items

17. ANS:

There are 2 employees who speak all three of the languages.

[pic]

REF: Applications OBJ: Section 5.1 LOC: CP1.01, CP1.02

TOP: Venn diagrams

18. ANS:

9C3 × 9C2 × 2C2 = 6048

There are 6048 possible line-ups.

REF: Applications OBJ: Section 5.2 LOC: CP1.02, CP1.04, CP1.06

TOP: Applying combinations

19. ANS:

5 × 3 × 7 – 1 = 104

You can make 104 different sums of money.

REF: Applications OBJ: Section 5.3 LOC: CP1.02, CP1.06

TOP: Applying combinations

20. ANS:

x 6 + 6x 5y + 15x 4y 2 + 20x 3y 3 + 15x 2y 4 + 6xy 5 + y 6

REF: Knowledge & Understanding OBJ: Section 5.4 LOC: CP1.07

TOP: Binomial theorem

PROBLEM

21. ANS:

There must be three each of the blue, gold, and white tiles. Applying the formula for permutations with some identical items shows that there are [pic] possible patterns.

REF: Communication, Thinking/Inquiry/Problem Solving OBJ: Section 4.3

LOC: CP1.05, CP1.08 TOP: Permutations with some identical items

22. ANS:

The number of students who participated in at least one of these extra-curricular activities is 190 – 10 = 180. Let n be the number of graduating students who participated in all three kinds of activities. Applying the principle of inclusion and exclusion,

n = 180 –111 – 26 – 67 + 13 + 9 + 11

= 9

So, 9 students were members of a team as well as the students’ council and a club.

[pic]

REF: Applications OBJ: Section 5.1 LOC: CP1.01, CP1.02

TOP: Venn diagrams

23. ANS:

Direct Method

The committee could include one, two, or three weightlifters. For each of these cases multiply the number of ways of choosing the weightlifters by the number of ways of choosing the remaining members of the committee. Add up the three cases to find the total number of possible committees:

5C1 × 13C2 + 5C2 × 13C1 + 5C3 × 13C0 = 390 + 130 + 10

= 530

There are 530 ways to choose a committee with at least one weightlifter on it.

Indirect Method

Find the total number of possible committees and subtract the number that do not include any weightlifters:

18C3 – 13C3 = 530

REF: Applications, Communication OBJ: Section 5.3 LOC: CP1.02, CP1.06, CP1.08

TOP: Applying combinations

24. ANS:

Direct Method

There are 6 pieces for piano and 12 for the other instruments. The students can choose 1, 2, 3, or 4 piano pieces. Consider each of these cases in turn.

1 piano piece: The students can choose the piano piece in C(6, 1) ways and the remaining 3 pieces in C(12, 3) ways. The number of combinations with 1 piano piece is C(6, 1) × C(12, 3) = 1320.

2 piano pieces: The number of combinations is C(6, 2) × C(12, 2) = 990.

3 piano pieces: The number of combinations is C(6, 3) × C(12, 1) = 240.

4 piano pieces: The number of combinations is C(6, 4) × C(12, 0) = 15.

The number of combinations that include at least 1 piano piece is the total of these four cases, 2565.

Indirect Method

Find the total number of possible combinations and subtract those that do not have any piano pieces:

C(18, 4) – C(12, 4) = 3060 – 495

= 2565

REF: Applications, Communication OBJ: Section 5.3 LOC: CP1.02, CP1.06, CP1.08

TOP: Applying combinations

25. ANS:

(ax + by)8 = a8x8 + 8a7x7by + 28a6x6b2y2 +...

a8 = 256

a = 2 or a = –2

|For a = 2, |For a = –2, |

|8a7b = 6144 |8a7b = 6144 |

|8(2)7b = 6144 |8(–2)7b = 6144 |

|b = 6 |b = – 6 |

REF: Applications OBJ: Section 5.4 LOC: CP1.07 TOP: Binomial theorem

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