Unit on Permutations and Combinations (Counting Techniques)

Transcription

1 Page 1 of 15 (Edit by Y.M. LIU) Page 2 of 15 (Edit by Y.M. LIU) Unit on Permutations and Combinations (Counting Techniques) e.g. How many different license plates can be made that consist of three digits followed by three letters? ( ) Fundamental Counting Principle In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so on, until nth has k n, the total number of possibilities of the sequence will be k k k e.g. A paint manufacturer wishes to manufacture several different paints. The categories include Colour: Red, Blue, White, Black, Green, Yellow Type: Latex, Oil Texture: Flat, Semigloss, High Gloss Use: Outdoor, Indoor How many different kinds of paint can be made if a person can select on colour, one type, one texture, and one use? (72) k n e.g. You bought five house numbers at Home Deport. The numbers are : 1, 2, 3, 4, and 5. (a) How many even three-digit house numbers can you from using these numbers? (24) (b) How many odd three-digit house numbers can you from using these numbers? (36) (c) How many of three-digit house numbers are greater than 300? (36) e.g. There are four blood types, A, B, AB, and O. Blood can also be Rh+ and Rh-. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled? (16) e.g. The province of New Brunswick uses a six-character license plate, three letters followed by three digits, with the restriction that the first letter must be a B. How many different license plates are possible? (676000) e.g. John is planning to drive from Vancouver to Winnipeg via Calgary. There are three roads from Vancouver to Calgary and two roads from Calgary to Winnipeg. How many different round-trip routs are there from Vancouver to Winnipeg, passing through Calgary, if no road is used more than once? (12) e.g. How many permutations can be formed using all the letters of the word FRUITAGE? (40320)

2 Page 3 of 15 (Edit by Y.M. LIU) Page 4 of 15 (Edit by Y.M. LIU) Factorial Notation For any counting n, where n is a positive integer and n 1 e.g. Solve for n. ( n 1)! (a) 7 n ( n 1) ( n 2) ! 1 e.g. Express each as a product in reduced form and its final answer. (a) 4! ( n 1)! (b) 30 ( n 3)! (b) 7! 3! (c) 8! 4!3! ( n 2)! (d) ( n 2)! (c) 20 ( n 1)! (e) ( n 3)! ( n 2)! (f) ( n 1)! (d) 42 ( n 2)! ( n 4)! (g) ( n 2)!

3 Page 5 of 15 (Edit by Y.M. LIU) Page 6 of 15 (Edit by Y.M. LIU) Permutation The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at a time. The is written as P, and the formula is n r Grouped Permutations In general, if objects are to be placed together, we can find the count by taking the product of the factorial of the number of groups and the factorial of each group. 6! 6! e.g. 6 P (6 4)! 2! P r n ( n r)! Linear Permutations e.g. A language teacher wants to keep books of the same language together on his shelf. If he as 12 spaces for five Chinese, 4 Japanese, and 3 French books, in how many way can be place them on the shelf? 3!5!4!3! e.g. In how many ways can 3 desks be filled from amongst 10 students? (750) e.g. How many different ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available? (42) e.g. Five people, A, B, C, D, and E, are seated on a bench. In how many ways can they be seated if: (a) there is no restriction. (120) e.g. There are 10 different books. How many ways can 4 of these books be arranged on a shelf? (5040) (b) A and B wish to be seated together. (48) e.g. John and Tom invited four other people to sit on their bench. In how many ways can these six people be seated on this bench if: (a) there are no restrictions. (720) (c) A and B must not sit together. (72) (b) John is seated at the left and Tom is seated at the right end. (24) (d) A and C sit together, and B and E sit together also. (24) e.g. How many 3-letter permutations can be formed from the letters of the word CLARINET? (336)

4 Page 7 of 15 (Edit by Y.M. LIU) Page 8 of 15 (Edit by Y.M. LIU) Circular Permutations (Not on the test) A circular permutation does not have a first or last position. The positions in the circle are relative to the other objects of this circle. Therefore, we must assign an object into the circle first to start, then the rest of the objects will have the same number of choices as if they were placed in a linear permutation. The number of permutation of n object in a circle with no restriction is (n 1)!. e.g. In how many ways can four people stand in (a) a line. (24) Permutations of n Objects When Some Are Alike: The number of permutations of n objects of which a objects are alike, another b objects are alike, another c objects are alike, and so on, is a! b! c!... e.g. How many different words can be formed using the letters of the word: a) SEA? (6) (b) a circle. (6) b) SEEM? (12) e.g. In how many ways can five people be seated around a circular table? (24) e.g. How many different words can be formed using the letters of the word MISSISSIPPI? (34650) e.g. In how many ways can five people, A, B, C, D, and D, be seated around a table if: (a) A and B must sit next to each other. (12) e.g. 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? (1260) (b) A and B must not be next to each other. (12) (c) A and B, C and D must be together. (8) e.g. A lacrosse term s record over a season was 15 wins, 4 losses, and 2 ties. a) In how many orders could this record have occurred? (813960) b) If you know that the term started the season strongly with five straight wins, how many orders are possible for the team s results? (120120)

5 Page 9 of 15 (Edit by Y.M. LIU) Page 10 of 15 (Edit by Y.M. LIU) e.g. Student A wants to visit student B. Roads are shown as lines on a grid. Only south and east travel directions can be used. The trip shown is described by the direction of each part of the trip: ESSESE How many different paths can A take to get to B? (20) Combination: A selection of distinct objects without regard to order is called a combination, which means order is not important in the selecting process. e.g. Given the letters A, B, C, and D, list the permutations and combinations for selecting two letters. e.g. There are 3 blue flags, 3 white flags, and 2 flags. How many different signals can be constructed by making a vertical display of 8 flags? (560) Combinations of r Objects Taken From n Distinct Objects: e.g. On a 5-question true-false test, two answers are T and three answers are F. How many different answer keys are possible? (10) e.g. On each grid, how many different paths are there from A to B? The notation nc r is used for the number of combinations of r objects taken from n distinct objects. n Pr ncr r! ( n r)! r! ( n r)! r! (a) (b) 16! 16 P3 (16 3)! 16! e.g. 16C ! 3! (16 3)!3! e.g. Determine the number of possible lottery tickets that can be created in 6/49 lottery where each ticket has six different numbers, in no particular order, chosen from the numbers 1 through 49 inclusive. ( )

6 Page 11 of 15 (Edit by Y.M. LIU) Page 12 of 15 (Edit by Y.M. LIU) e.g. According to the Guinness Book of Records, Adam Borntrager s family is the largest in the world. Mr. Borntrager and his wife have 11 children, 115 grandchildren, 529 greatgrandchildren, and 20 great-great-grandchildren. How many handshakes would take place if every family member at the reunion shook every other family member s hand? (228826) e.g. In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? e.g. A bicycle shop owner has 12 mountain bicycles in the showroom. The owner wishes to select 5 of them to display at a bicycle show. How many different ways can a group of 5 be selected? (792) e.g. A standard deck of 52 playing cards consists of 4 suits (spades, hearts, diamonds, and clubs) of 13 cards each. a) How many different 5-card hands can be formed? ( ) e.g. A group of five students is to be selected from a class of 35 students. b) How many different 5-card red hands can be formed? (65780) a) How many different groups can be selected? (324632) c) How many different 5-card hands can be formed containing at least 3 black cards? ( ) b) Lisa, Gwen, and Al are students in this class. How many of the possible groups include all three of these students? (496) c) How many groups do not include all three of these students? (324136) e.g. Suppose there are 5 books (A, B, C, D, and E) that are to be placed in 3 available positions on a shelf. How many different arrangements are possible? (60) e.g. How many different committees of 3 people can be selected from 8 people? (56) e.g. From 5 books (A, B, C, D, and E), how many reading lists of 3 different books can be made up? (10)

7 Page 13 of 15 (Edit by Y.M. LIU) Page 14 of 15 (Edit by Y.M. LIU) Pascal s Triangle: The Binomial Theorem (using combinations): In the (n + 1) th row, the (r +1) th number is the number of combinations of n objects taken r at a time, n C r. r!( n r)! e.g. In the 7 th row, the 4 th number is the number of combinations of 6 objects taken 3 at a 6! time, 6C !3! e.g. What is the second number in the 50 th row? (49) The expansion of ( a b ) n, where n is a natural number, is given by ( ab) n C a n Ca n b C a n b C a n b C a n k b k C ab n C b n n 0 n 1 n 2 n 3 n k n n1 n n nk k The general term is of the form tk 1 ncka b, where k = 0 gives the first term, k = 1 gives the second term, k = 2 gives the third term, and so on. Therefore, n n nk k n k k 0 ( a b) C a b The Binomial Theorem (using algebraic expressions): nn ( 1) nn ( 1)( n2) nn ( 1)( n2) [ n( k1)] ( ab) a na b a b a b a b b 2! 3! k! n n n1 n2 2 n3 3 nk k n n is a whole number. e.g. Expand, then simplify 3 ( x 3) e.g. Determine the number of pathways from A to B in the following street arranges. A B e.g. Determine the 7 th term in the expansion of 10 ( x 2). A e.g. Determine the fifth term in the expansion of 7 (2x 5). B

8 Page 15 of 15 (Edit by Y.M. LIU) 20 e.g. Write the first four terms of the binomial expansion of (1 x). e.g. Use the Binomial Theorem to expand 5 ( x 3). 3 2 e.g. In the expansion of x 2 write and simplify: x (a) The fifth term 10 (b) The term containing 15 x (c) The constant term