PERMUTATIONS AND COMBINATIONS 1. Fundamental Counting Principle Assignment: Workbook: pg. 375 378 #1-14 2. Permutations and Factorial Notation Assignment: Workbook pg. 382-384 #1-13, pg. 526 of text #22 3. Permutations with Repetitions and Restrictions Assignment: Workbook: pg. 388 391 #1-15, pg. 525 of text #8 4. Combinations pg. 528-536 Assignment: Workbook: pg. 396 398 #1-12, pg. 402-404 #1-16, pg. 534 of text #6 5. The Binomial Theorem Assignment: Workbook: pg. 410 412 #4, 7-13, pg. 415 #1abcdef 6. Chapter Quiz 7. Chapter Review pg. 546-547 Assignment: pg. 546-547 #1-12, 14-18 8. Chapter Exam
FUNDAMENTAL COUNTING PRINCIPLE Learning Outcomes: To solve problems using the fundamental counting principle To determine, using a variety of strategies, the number of permutations of n elements taken r at a time. Ex. A toy manufacturer makes a wooden toy in three parts: Part 1: the top part may be coloured red, white or blue Part 2: the middle part may be orange or black, Part 3: the bottom part may be yellow, green, pink, or purple How many different parts are there to the toy? For each part of the toy, how many choices of colours do you have? Multiply all the choices together, now you have the total amount of possible toys:.. = top middle bottom coloured toys Check your answer using a tree diagram: Make a conjecture about how you can use multiplication only to arrive at the number of different coloured toys possible. 2
The Fundamental Counting Principle Consider a task made up of several stages. If the number of choices for the first stage is a, the number of choices for the second stage is b, the number of choices for the third stage is c, etc., then the number of ways in which a task can be completed is a x b x c x.. This is called the fundamental counting principle. Ex. Determine the number of distinguishable four letter arrangements that can be formed from the word ENGLISH if: a. letters can be repeated? b. no letters are repeated and: i) there are no further restrictions? ii) the first letter must be E? iii) the word must contain G? iv) the first and last letters must be vowels? Ex. The telephone numbers allocated to subscribers in a rural area consist of one of the following: - the digits 345 followed by any three further digits or - the digit 2 followed by one of the digits 1 to 5, followed by any three further digits. How many different telephone numbers are possible? 3
Ex. Car number plates in an African country consist of a letter other than I or O followed by three digits, the first of which cannot be zero, followed by any two letters which are not repeated. How many different car number plates can be produced? Ex. Consider the digits 2, 3, 5, 6, 7 and 9. a) If repetitions are not permitted, how many 3-digit number can be formed? b) How many of these are: i) less than 400? ii) even? iii) multiples of 5? Workbook: pg. 375 378 #1-14 4
Learning Outcomes: PERMUTATIONS AND FACTORIAL NOTATION To solve problems using the fundamental counting principle To determine, using a variety of strategies, the number of permutations of n elements taken r at a time. To solve an equation that involves n P r notation Factorial Notation Consider how many ways there are of arranging 6 different books side by side on a shelf. In this example we have to calculate the product 6x5x4x3x2x1. In mathematics this product is denoted by 6! ( factorial or factorial 6 ) In general n!=n(n-1)(n-2)(n-3).(3)(2)(1), where n W Ex. Use the factorial key on your calculator to solve 6! Ex. To simplify 10!, there are several approaches: 7! a.use you calculator: b. By Cancellation: 10! 7! Ex. Find the value of Ex. Simplify the following expressions: a. ( ) b. ( ) 5
Permutations An arrangement of a set of objects in which the order of the objects is important is called a permutation. Ex. How many permutations are there of the letters of the word: a. REGINA: b. KELOWNA: The number of permutations of n different objects taken r at a time is: n P r n! ( n r )! Ex. Use the n P r key on your calculator to evaluate 8 P 3. Then solve using factorials. 8: Math PRB 2 3 336 Using Factorials: n p r n! 8! 8! 8 7 65! 8 7 6 336 ( nr)! (8 3)! 5! 5! Defining 0! If we replace r by n in the above formula we get the number of permutations of n objects taken n at a time. This we know is n! n p n n! n! n! ( n n)! 0! For this to be equal to n! the value of 0! Must be 1. 0! Is defined to have a value of 1 Try it: Solve 6
Ex. In a South American country, vehicle license plates consist of any 2 different letters followed by 4 different digits. Find how many different license plates are possible using: fundamental counting principle: permutations: (letters)(digits) Ex. Solve for n in the equation np4 28 n1 P2 n! 28( n1)! ( n 4)! ( n1) 2! n! 28 n 1! ( n4)! ( n3)! In many cases involving simple permutations, the fundamental counting principle can be used in place of the permutation formulas. Assignment: pg. 382-384 #1-13 7
Learning Outcomes: PERMUTATIONS WITH RESTRICTIONS AND REPETITIONS To solve problems using the fundamental counting principle To determine, using a variety of strategies, the number of permutations of n elements taken r at a time. To solve an equation that involves n P r notation To solve counting problems when two or more elements are identical Permutations with Restrictions In many problems restrictions are placed on the order in which objects are arranged. In this type of situation deal with the restrictions first. Ex. In how many ways can all of the letters of the word ORANGES be arranged if: a.)there are no further restrictions: b.)the first letter must be an N? : c.)the vowels must be together in the order O, A, and E: You need to multiply by 5 because the OAE can fill into any of the 5 slots 8
Ex. Find the number of permutations of the letters in the word KITCHEN if: a.)the letters K, C, and N must be together but not necessarily in that order b.)the vowels must not be together easier to find complementary event: vowels kept together, then subtract from no restrictions (7!) 7! - 2. 1. 5. 4. 3. 2. 1 vowels =7! ((2!)(5!)(6)) = 5040 1440 = 3600 Ex. In how many different ways can 3 girls and 4 boys be arranged in a row if no two people of the same gender can sit together Permutations with Repetitions The following formula gives the number of permutations when there are repetitions: The number of permutations of n objects, where a are the same of one type, b are the same of another type, and c are the same of yet another type, can be represented by the expression below n! abc!!! 9
Ex. Find the number of permutations of the letters of the word: a.) VANCOUVER: b.) MATHEMATICAL: Ex. How many arrangements of the word POPPIES can be made under each of the following conditions? a.) without restrictions: b.) if each arrangement begins with a P: P OPPIES The first letter is a P, and the next 6 letters can be in any arrangement (remember there is a repetition of 2P s within the remaining 6 letters) c.) if all the P s are together: PPP OIES 1 2 3 4 5 d.) if the first letter is P and the next one is not P: 10
Ex. Brett bought a carton containing 10 mini boxes of cereal. There are 3 boxes of Corn Flakes, 2 boxes of Rice Krispies, 1box of Coco Pops, 1 box of Shreddies, and the remainder are Raisin Bran. Over a ten day period Brett plans to eat the contents of one box of cereal each morning. How many different order are possible if on the first day he has Raisin Bran? CF, 2 RK, 1 CP, 1 S, 3 RB (will only have 2 after first day) Consider the following problem: Ex. A city centre has a rectangular road system with 5 streets running north to south and 6 avenues running west to east. a.draw a grid to represent this situation Sean b. Sean is driving a car and is situated at the extreme northwest corner of the city centre. In how many ways can he drive to the extreme southeast corner if at each turn he moves closer to his destination (assume all streets and avenues allow two way traffic) 11
Ex. Find the number of pathways from A to B if paths must always move closer to B. A One possible path is shown, any path from A to B must travel 3 right, 2 down and 1 forward B Find the number of pathways from A to B if paths must always move closer to B A B Homework: Workbook: pg. 388 391 #1-15 12
LESSON 4: COMBINATIONS Learning Outcomes: To explain the differences between a permutation and a combination To determine the number of ways to select r elements from n different elements To solve problems using the number of combinations of n different elements taken r at a time To solve an equation that involves n C r notation When is order not important? 1. From a group of four students, three are to be elected to an executive committee with a specific position. The positions are as follows: 1 st position President 2 nd position Vice President 3 rd position Treasurer a. Does the order in which the students are elected matter? Why? b. In how many ways can the positions be filled from this group? 2. Now suppose that the just elected three students are to be selected to serve on a committee. a. If all the committee positions are the same, does it matter if you are selected first or third? b. Is the order in which the four students are selected still important? Why or why not? 13
c. How many committees from the original group of four students are now possible? 4 1 (President, VP, Treasurer), 2(President, VP, student 4), 3( VP, Treasurer, student 4), 4( President, Treasure, student 4) 3. You are part of a group of 6 students who are about to shake hands with every other person. a. How many students shake hands at one time? b. Does the order of the handshakes matter? c. How many handshakes are possible if each student shakes every other student s hand once? 4. Part 1 deals with permutations, part 2 and 3 dealt with combinations. What is the biggest difference between a permutation and a combination? Permutations require that the order of the grouped objects is important. Combinations involve arrangements where order of the grouped objects is not important. A combination is a selection of a group of objects, taken from a larger group for which the kinds of objects selected is important, but not the order in which they are selected. There are several ways to find the number of possible combination. One is to use reasoning. Use the fundamental counting principle and divide by the number of ways that the object can be arranged among themselves. 14
For example, calculate the number of combinations of three digits made from the digits 1, 2, 3, 4, and 5: 5 x 4 x 3 = 60 However, 3 digits can be arranged 3! ways among themselves. So: Formula: The number of combinations of n items take r at a time is: n n! C r n r!! r Use the formula to solve the last problem: ( ) The n C r key on the calculator can be used to evaluate combinations: Math PRB 3 ncr In some texts n C r is written as n r Ex. Three students from a class of 10 are to be chosen to go on a school trip. In how many ways can they be selected? Ex. To win the LOTTO 649 a person must correctly choose six numbers from 1 to 49. Jasper, wanting to play LOTTO 649, began to wonder how many numbers he could make up. How many choices would Jasper have to make to ensure he had the six winning numbers? 15
Ex. The Athletic Council decides to form a sub-committee of seven council members to look at how funds raised should be spent on sports activities in the school. There are a total of 15 athletic council members, 9 males and 6 females. The sub-committee must consist of exactly 3 females. a. In how many ways can the females be chosen? b. In how many ways can the males be chosen? c. In how many ways can the sub-committee be chosen? Exactly 3 females (females)(males) d. In how many ways can the sub-committee be chosen if Bruce the football coach must be included? (females)(males)(bruce) Ex. Consider a standard deck of 52 cards. How many different five card hands can be formed containing: a. at least 1 red card find complementary even ( no restrictions no red) C C 52 5 26 5 = 2 533 180 different five card hands 16
b. at most 2 kings 3 events 0 kings, 1 king, 2 kings c. exactly two pairs? 1 st pair x 2 nd pair x 1 other card x what number for the pair Combinations which are equivalent Ex. Jane calculated 10 C 2 to be 45 arrangements. She then calculated 10 C 8 to be 45 arrangements. Use factorial notation to prove this: C C 10 2 10 8 10! 2!8! 10! 8!2! 1098! 1098! 2!8! 8!2! 45 = 45 17
Solving for n in Combinations Problems Ex. During a Pee Wee hockey tryout, all the players met on the ice after the last practice and shook hands with each other. How many players attended the tryouts if there were 300 handshakes in all? A handshake requires 2 people C n 2 300 n! 300 ( n 2)!2! n! 600 ( n 2)! Polygons and Diagonals The number of diagonals in a regular n-sided polygon is: nc n 2 Ex. How many diagonals are there in a regular octagon? Ex. A polygon has 65 diagonals. How many sides does it have? C n 2 n 65 n! 65 n ( n 2)!2! n( n 1)( n 2)! 2(65 n) ( n 2)! Workbook: pg. 396 398 #1-12, pg. 402-404 #1-1 18
LESSON 5: THE BINOMIAL THEOREM Learning Outcomes: To relate the coefficients in the expansion of ( ), n ε N, to Pascal s triangle and to combinations To expand ( ), n ε N, in a variety of ways, including the binomial theorem To determine a specific term in the expansion of ( ) Pascal s Triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Notice the pattern? Can you continue the pattern for the next two rows? Recall Pathway questions, can you solve the following question using pascal s triangle? Find the number of pathways from A to B if paths must always move closer to B A B 19
How about this one? Ex. A city centre has a rectangular road system with 5 streets running north to south and 6 avenues running west to east. a.draw a grid to represent this situation Sean Ex. Complete the following expansions: a. ( ) b. ( ) c. ( ) Notice that the amount of terms is always 1 greater than the exponent. How do the coefficients of the simplified terms in your binomial expansions relate to Pascal s Triangle? The coefficients related to the row of pascal s. 20
Binomial Theorem 1 2 2 n n n n n k k n x y nc0x nc1x y nc2 x y... nck x y... ncn y, where ni, n 0 All binomial expressions will be written in descending order of the exponent of the first term in the binomial. The following are some important observations about the expansion of x y represent the terms of the binomial and nεn: The expansion contains n + 1 terms The sum of the exponents in any term of the expansion is n. Expand ( ) using the binomial theorem: How many terms will we expect? n, where x and y List the values of n, x and y: Expand using the formula: x = x, n = 4, y = y 1 2 2 n n n n n k k n x y nc0x nc1x y nc2 x y... nck x y... ncn y Ex. Expand 3x 2 3 21
General Term of the Expansion of ( x y ) n The term n C x k nk y k is called the general term of the expansion. It is the k 1 th term in the expansion (not term k) t C xnk yk t1 n k Note: expansion starts at a k value of 0, so the k value is one less than term requested Ex. a) Find the fifth term of 8 x y Fifth Term, think about the expansion: Dealing with the x values: Dealing with the y values: Notice after 5 terms our exponent is 4 Notice after 5 terms our exponent is 4 9 terms, n=8, k = 4 nk k tk 1 nck x y 4 4 t C x y 41 8 4 t5 70x y 4 4 b. the middle term of 2x 5 6 7 terms, n = 6, k = 3 (middle term would be the 4 th term) 22
Ex. One term in the expansion of 10 numerical value of a. x a is 3 281 250 4 x. Determine the What term will be This ensures the x values will cancel out because they will have the same exponent. n nk k C x y 3281250x k C x y 3281250x 106 6 4 10 6 4 23
Ex. Find the constant term (the term independent of x) in the expansion of ( ) To have a constant term, what should the exponent of x be? General Strategy: 1. Put equation into general term formula 2. Simplify/ expand formula 3. Solve for exponent of variables (gives k value) Break down the equation to get x by itself: Rewrite equation : ( ( ) ), n = 15, k = k ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) Use general term formula: ( ) ( ) ( )( ) ( ) Workbook: pg. 410 412 #4, 7-13, pg. 415 #1abcdefg 24