Sec. 4.2: Introducing Permutations and Factorial notation

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1 Sec. 4.2: Introducing Permutations and Factorial notation Permutations: The # of ways distinguishable objects can be arranged, where the order of the objects is important! **An arrangement of objects in a specific order. ORDER IS IMPORTANT! Ex#1 List all the different permutations of the letters AB Ex#2 List all the different permutations of the letters ABC Ex#3 List all the different permutations of the letters ABCD 1

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3 How many permutations can be formed using all the letters of the word MARCH A map of Canada is to be coloured with a different colour for each province and territory. If 13 colours are available, in how many ways can the map be coloured that look different from each other? (no repetition of colours) A baseball team has 9 players on the field. In how many ways can 9 players be arranged on the field? A family of 6 is lining up for a picture. In how many different ways can the members of the family line up for the picture 3

4 Practice Questions from this section: page 243 #1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 16 4

5 Sec. 4.3: Permutations when all objects are distinguishable (no repetition of objects) List all permutations of letters in word MATH taking 2 letters at a time. Permutation Formula n = total # of objects r= # of objects being selected and arranged npr = # of permutation 5

6 How many 4 letter permutations of word MATH How many 2 letter permutations of word MATH 6

7 EXAMPLES: 1. A disc jockey has a total of 7 songs to play. How many different ways can only 4 songs be arranged on the radio with no repetition? 7

8 EXAMPLES: 2. 6 sprinters are in the final race. How many different ways are there to award the gold, silver, and bronze medals? 8

9 EXAMPLES: 3. How many ways can a president, vice president and secretary be chosen from a 11 member committee? 9

10 4. How many different ways can 5 raffle tickets be selected from 100 tickets if each tickets wins a different prize? 5. How many ways can Laura colour 6 provinces on a map if she has a set of 12 coloured pencils? 6. How many ways can Shelby pick the lead actor, supporting actor, background actor and stage hand from 15 auditioning actors? 10

11 Solving a permutation problem involving cases Ex page 250 The password can use any digits form 0 to 9 and or an y letters of the alphabet. The password is case sensitive, so she can use both lower and upper case letters. A password must be at least 5 characters to a maximum of 7 characters, and each character can be used only once in the password. How many different passwords are possible? Suppose a online magazine requires each subscriber to have a password with exactly 8 characters, using the same requirements for characters. Which is more secure? 11

12 Solving a permutation problem involving cases Ex page 250 The password can use any digits form 0 to 9 and or an y letters of the alphabet. The password is case sensitive, so she can use both lower and upper case letters. A password must be at least 5 characters to a maximum of 7 characters, and each character can be used only once in the password. How many different passwords are possible? Suppose a online magazine requires each subscriber to have a password with exactly 8 characters, using the same requirements for characters. Which is more secure? 12

13 CALCULATE THE FOLLOWING 13

14 Kelly is the team captain of a 15 member soccer team. How many ways can Kelly and two other players line up to receive the championship trophy, if the captain must be first in line? Ex # 5 page 252 A social insurance number (SIN) in Canada consists of a nine digit number that uses digits 0 to 9. If there are no restrictions on the digits selected for each position in the number, how many SINs can be created if each digit can be repeated? How does this compare with the number of SINs that can be created if no repetition is allowed. Your turn page 253 a) How many nine digit SINs do not start with 0, 8 and 9? b) SINs starting with the digit 9 are issued to temporary residents. How many SINs are there in total? 14

15 Review How many ways can John, Adam, Beth, Charlie and Sue be seated in a row if: a) Charlie must be in the second chair? b) Boys and girls alternate, with a boy starting the line? c) A boy must be on both ends. d) Sue must be in the second position and a boy must be in the third position. How many ways can you order letters in the word KITCHEN if it must start with a consonant and end with a vowel? How many two letter, three letter and four letter arrangements can be formed from the word PENCIL? 15

16 Permutation Practice Questions Practice Questions from this section: page 255 1a, e, f, 2b, 4, 5, 6, 7, 8, 9, 10abc, 12, 13 16

17 4.4: Permutations involving Identical objects (repetition of objects allowed) Permutations are less when letters are repeated. 17

18 Examples: A teacher has 2 exams to be administered to a class of 20. the teacher has 12 of one type and 8 of another type. How many different ways can the exams be given to the students? 18

19 1. How many permutations are there of all the letters in the word BOOKKEEPER? 2. How many permutations are there of all the letters in the word PARALLEL? 3. A builder has three models of homes from which customers can choose, A, B, and C. On one side of a street, the builder sold three model A homes, four model B homes, and two model C homes. In how many ways can the homes be arranged along the street? 4. Kathryn s soccer team played a good season, finishing with 16 wins, 3 losses, and 1 tie. In how many orders could these results have happened? 5. Six friends shared a bag of assorted doughnuts. In how many ways ca the friends share the doughnuts if the bag contains a. 6 different doughnuts? b. 4 plain doughnuts and 2 Boston cream doughnuts? c. 2 plain doughnuts, 2 Boston cream doughnuts, and 2 jelly doughnuts? 19

20 6. A multiple choice test has 10 questions of which 1 is answered A, 4 are answered B and 3 are answered C and 2 are answered D. How many different answer keys are possible? 7. How many ways can the letters of the word CANADA be arranged, if the first letter must be N and the last letter must be C? b) How many different arrangements are there for the six letters if there are no conditions for where letters must be placed? c) How many different arrangements, if the first letter has to be a C. 8. Eight coloured flags are arranged on a pole (3 red, 2 blue and 3 orange). a) How many ways can they be arranged, if the pole of flags begins with blue and ends with blue? b) How many ways can they be arrange, if the pole of flags starts with a blue and ends with an orange? 9. How many ways can you arrange the letters from the word TREES if: a) It begins with an E? b) It begins with exactly one E? c) consonants and vowels alternate? d) It must start with a consonant and end with a vowel? 20

21 PATHWAY PROBLEMS A tractor is traveling from Point A to point B. The tractor can only make south and east moves only. How many possible routes are there 21

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23 4.5 Combinations A combination is a selection of objects where the order is NOT IMPORTANT. Ex. the two objects A and B have one combination because AB is the same as BA. Key words in word problem to indicate it is a combination question include: GROUP CHOSEN SELECTION 23

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Determine the number of permutations of n objects taken r at a time, where 0 # r # n. Holly Adams Bill Mathews Peter Prevc

Determine the number of permutations of n objects taken r at a time, where 0 # r # n. Holly Adams Bill Mathews Peter Prevc 4.3 Permutations When All Objects Are Distinguishable YOU WILL NEED calculator standard deck of playing cards EXPLORE How many three-letter permutations can you make with the letters in the word MATH?