4.1 Organized Counting McGraw-Hill Ryerson Mathematics of Data Management, pp

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1 Name 4.1 Organized Counting McGraw-Hill yerson Mathematics of Data Management, pp Draw a tree diagram to illustrate the possible travel itineraries for Pietro if he can travel from home to Ottawa by bus, car, or train, and then from Ottawa to Orlando, Florida by bus, train, or plane. 2. A new computer game has two possibilities, the gate is either or. The characters must meet five gates during their time allotment. Draw a tree diagram to illustrate the possible scenarios for a player. 3. A physical education teacher has five pairs of running shoes, eight pairs of sweat pants, and 12 T-shirts. How many different outfits cans she wear? 4. A code consists of three letters followed by two numbers. No repetition is allowed. How many different codes are available? 5. A drama club has 15 members. In how many ways can a president, vice-president, and secretary be chosen? 6. In how many ways can four boys and three girls be seated along one side of a table? 7. There are two bus routes from Lindsay s house to the shopping centre, four bus routes from the shopping centre to downtown, and three bus routes from downtown to Lindsay s house. Draw a tree diagram to illustrate the possible routes Lindsay can take if she leaves her house on Saturday morning to go to the shopping centre, and she wants to stop downtown on her way home. 8. ridget works part time in a shoe store. Sometimes when it is not busy, she rearranges the shoes for fun. If she takes six different pairs of shoes and rearranges them in a row, in how many ways can she rearrange them so that no two shoes match? 9. John is having a party and he wants to seat his guests at his dining room table along one side. In how many ways can John seat himself and his six guests? 10. There are three movies playing in town. Four of the high schools are holding dances. Alex and Josie want to go out for the evening, first to a movie and then, to a dance. How many choices are there for Alex and Josie? 11. Jing Jing works in a store and she spends some of her time organizing the stationary section. If there are nine different items on the rack, in how many ways can she organize the rack so that the two most expensive items are not together? 12. There are seven members in the track team. If all of them are to run in a heat, how many different top three finishes could there be? 13. A committee of three is to be formed from five math teachers and four English teachers. In how many ways can the committee be formed if order matters and there a) are no restrictions? b) must be one mathematics teacher? c) must be at least one mathematics teacher? d) must be only mathematics teachers? Copyright 2003 McGraw-Hill yerson Limited Chapter 4 MH 45

2 Name 4.2 Factorials and Permutations McGraw-Hill yerson Mathematics of Data Management, pp Mrs. Edwards has to mark 33 quizzes tonight. In how many different orders can she mark them? 2. In how many ways can you choose a president and vice-president from a group of 11 people? 3. A quarterback has a series of six plays possible. If the coach asks the quarterback not to repeat any plays in a game, how many different orders of plays is possible? 4. Assuming that everyone in a particular school has three initials, find out what is the smallest number of students in a school for which there must be at least two with the same initials. 5. Use a calculator to find each of the following: a) 5P 2 b) 10 P 10 c) 8P 1 d) 15 P 7 6. In how many ways can five girls and five boys in a choir stand in a line if boys and girls must alternate positions? 7. The school each team is attending an invitational tournament, but can only take the top five students. In how many ways can you choose the top five students from the full complement of 11 students on the school each team? Assume that the order in which the students are chosen matters. 8. a) Explain what is meant by the term factorial. b) Give an example of a situation in which you might need to calculate 4!. 9. The junior choir and the senior choir are both singing selections at the holiday concert. The junior choir has a repertoire of four songs, and the senior choir can sing any of eight songs. In how many different ways can the junior choir sing two selections and the senior choir three selections? Assume the junior choir always sings first. 10. The top two students in the grade 12 mathematics' class are to receive awards for the highest mark and the second highest mark in the school. If there are seven students in the running before the final examination, in how many different orders could you choose the top two students? 11. How many ways can you pick a president, vice-president, and secretary from a group of six boys and five girls if a) there are no restrictions? b) there must be at least one boy chosen? c) there must be only one girl chosen? 46 MH Chapter 4 Copyright 2003 McGraw-Hill yerson Limited

3 Name 4.3 Permutations With Some Identical Elements McGraw-Hill yerson Mathematics of Data Management, pp In how many different ways can your arrange five flags in a row if there are two red flags and three blue flags? 2. How many distinct words can you make using all the letters of the word PAALLELEPIPED? 3. In how many ways can you arrange all the letters of the word TOONTO a) that begin with a T? b) that end in a T? c) that have both Ts together? 4. Find the number of ways of arranging all of the letters of the word TENNESSEE a) if there are no restrictions b) if the first two letters must be EE c) if the first two letters must not be EE 5. Laura s soccer team played a good season, finishing with 12 wins, four losses, and two ties. In how many orders could this have happened? Explain. 6. How many different 7-digit telephone numbers contain three 5s, two 2s and two 1s? 7. John is putting ceramic tile on the wall of the school store and he chooses three colours of tile: green, gold, and white. If John has purchased 20 tiles in total, five green, 10 gold, and five white, in how many ways can he arrange the tiles on the wall in a 4 5 pattern? 8. In how many ways can you seat four adults and seven children along one side of a rectangular table? Assume the adults are indistinguishable from each other and the children are indistinguishable from each other. 9. How many different words can you make using all the letters of the word MISSISSIPPI if a) there are no restrictions? b) the word must end with an I? c) the word cannot end in an I? 10. If there are 12 books on the shelf, in how many ways can your arrange them, based only on colour, so that a) each book is a different colour? b) two books are yellow, five are red, and the rest are of different colours? Copyright 2003 McGraw-Hill yerson Limited Chapter 4 MH 47

4 Name 4.4 Pascal's Triangle McGraw-Hill yerson Mathematics of Data Management, pp Fill in the missing numbers of this part of Pascal s triangle Find the missing numbers of this part of Pascal s triangle Determine the sum of the terms in the 10th row of Pascal s triangle. 4. Determine the sum of the 15th row of Pascal s triangle. 5. List the first eight triangular numbers. 6. State the 5th and 12th triangular numbers. 7. a) Find your own pattern in Pascal s triangle. b) Explain in words, what you have found. 8. Prove 2 n 2 n 1 = 2 n What is the sum of the entries in the 17th row of Pascal s triangle? 48 MH Chapter 4 Copyright 2003 McGraw-Hill yerson Limited

5 Name 4.5 Applying Pascal's Method McGraw-Hill yerson Mathematics of Data Management, pp The first four numbers in the row n = 90 of Pascal s triangle are 1, 90, 4005, and a) Write the first four terms in the expansion of (x + y) 90. b) Write the last three coefficients in the row n = 90 of Pascal s triangle. c) Write the last four terms of the expansion (x + y) The first four terms of the expansion of (x + y) 100 are x x 99 y x 98 y x 97 y 3. a) Write the first four coefficients in the row n = 100 of Pascal s triangle. b) Write the last four coefficients in the row n = 100 of Pascal s triangle. c) Write the last four terms in the expansion of (x + y) In the following arrays of letters, start at the top and proceed to the next row diagonally left or right. How many different paths will spell each word? a) P b) F A A E S S S C C C C M M M M A A A A A A L L T T c) S d) G I I A A E E E U U U S S P P P S I I I I N N N S S S S K K K I I 4. Determine the number of possible routes from X to Y if you travel only north or west. a) Y b) Y X X Copyright 2003 McGraw-Hill yerson Limited Chapter 4 MH 49

6 Practice/Assessment Masters Answers CHAPTE 4 Permutations and Organized Counting 4.1 Organized Counting 1. us Car Train us Train Plane us Train Plane us Train Plane Downtown 1 Shopping Centre 1 Shopping Centre Downtown 2 Downtown 3 Downtown 4 Downtown 1 Downtown 2 Downtown 3 Downtown a) 504 b) 180 c) 480 d) Factorials and Permutations 1. 33! a) 20 b) c) 8 d) Answers may vary a) 990 b) 930 c) Permutations With Some Identical Elements ! = !2!3!3! 3. a) 120 b) 120 c) a) 3780 b) 630 c) a) b) c) a) 12! b) Pascal's Triangle , 120, 252, 330, , 2300, 325, 351, , 3, 6, 10, 15, 21, 28, , Answers may vary. 8. Proofs may vary Copyright 2003 McGraw-Hill yerson Limited Chapter 4 MH 53

7 4.5 Applying Pascal's Method 1. a) x 90, 90x 89 y, 4005x 88 y 2, x 87 y 3 b) 4005, 90, 1 c) x 3 y 87, 4005x 2 y 88, 90xy 89, y a) 1, 100, 4950, b) , 4950, 100, 1 c) x 3 y 97, 4950x 2 y 98, 100xy 99, y a) 20 b) 20 c) 204 d) 6 4. a) 120 b) 252 Study Master, Chapter a) b) c) ! ~ ! a) 1, 22, 231, 1540, 7315 b) t 11, 5 = t 10, 4 + t 10, 5 t 50, 21 = t 49, 20 + t 49, 21 t n + 2, r + 2 = t n + 1, r t n + 1, r b) i) Uses multiplicative principle; each of the three signals area separate stage with three choices in each stage. ii) Uses both; each of 1, 2, 3 signals use the multiplicative principle and are mutually exclusive actions. 4. a) 840 b) 480 c) a) 24 b) 180 c) a) t n, 2 is the entry indicating the maximum number of intersection points of n straws. b) 10 Alternative Test, Chapter 4 1. a) 16 b) c) Answers may vary. For example, there are 4 stages and 2 choices in each stage, so there are 2 4 = 16 outcomes. 2. a) 18! 18! 15! = 306 b) = 336 c) = ! 5! 13! d) 11! 9! = = 1 0! 9! 3. a) i) 27 ii) MH Chapter 4 Copyright 2003 McGraw-Hill yerson Limited