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SESSION 4 Permutations Combinations Polynomials SESSION 4 Permutations Combinations Polynomials Updated April 2014 Updated April 2014 Mathematics 30-1 Learning Outcomes Permutations and Combinations General Outcome: Develop algebraic and numeric reasoning that involves combinatorics. Specific Outcome 1: Apply the fundamental counting principle to solve problems. - Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams. - Explain, using examples, why the total number of possible choices is found by multiplying rather than adding the number of ways the individual choices are made. - Solve a simple counting problem by applying the Fundamental Counting Principle. Fundamental Counting Principle If one task can be performed in a ways, a second task in b ways, and a third task in c way, and son on, then all of the tasks can be performed in ways. To arrange n objects, write down n blanks (spaces). Fill in each blank with the number of possible objects which could be place in that space. Then multiply. It restrictions have been placed on any of the blanks, ALWAYS deal with those restrictions (spaces) first. Updated April 2014 Example 1 – How many 3 letter words can be created, if a) No repetitions are allowed? b) If repetitions are allowed? Example 2 -- You are given the word SASKATOON. a) How many arrangements are there of all of the letters in the word SASKATOON? . b) How many arrangements are there of all the letters in the word SASKATOON, if the arrangement must start with an S? . Example 3 – Amy wants to make a sandwich. She can only pick ONE ITEM from EACH of the following categories: bread type, meat options and vegetable choice. Given the choices supplied below, determine the total number of options Amy has. Bread type – Wheat or Italian Meat options – turkey, ham, beef Vegetable choice – tomatoes, lettuce, cucumbers and pickles Updated April 2014 Example 4 – A dresser has four knobs. If you have 6 different colors of paint available, how many different ways can you paints the knobs? Example 5 – The number the distinguishable arrangements of the word KITCHEN, if the vowels must stay together, is A. B. C. 7P2 5P5 D. Updated April 2014 Specific Outcome 2: Determine the number of permutations of n elements taken r at a time to solve problems. - Understand permutation problems that involve repetition or like elements and constraints - Determine, in factorial notation, the number of permutations of n different elements taken n at time, to solve problems - Determine, using a variety of strategies, the number of permutations of n different elements, taken r at a time, to solve problems - Explain why n must be greater than or equal to r in the notation nPr - Solve equations that involve nPr notation, such as nP2 = 30 - Understand single 2-dimensional pathways can be used as an application of repetition of like elements - You are not expected to understand circular or ring permutations - You should understand handshakes questions Factorial notation is an abbreviation for products of successive positive integers. ( )( ) A permutation is an arrangement of objects in a definite order. The number of permutations of n different objects taken r at a time is given by the following formula. ( ) Deal with restrictions (constraints) first. A set of n objects containing a identical objects of one kind, b identical objects of another kind, and so on, can be arranged in ways. Some problems may have more than one case. When this happens, establish cases that, when taken together, cover all the possibilities. Calculate the number of arrangements for each case and then add the values of all cases to obtain the total number of arrangements. Updated April 2014 Example 1 -- To accessorize her outfit, Jane will choose 1 of 4 handbags, 1 of 5 hats, and 1 of 3 coats. How many different outfits can Jane create by changing these accessories? A. 3 B. 12 C. 60 D. 220 Example 2 -- How many different passwords can be made from all the letters in the word CALCULUS? A. 2500 B. 5040 C. 6720 D. 40320 1. If all the letters in the word PENCILS are used, the number of arrangements with all the vowels together is _____________? Example 3 -- What is the solution set for r given 7 Cr = 21? A. {2} B. {3} C. {2, 5} D. {3, 4} Example 4 – The number of three digit or four digit even numbers that can be formed from the numbers 2, 3, 5, 6 and 7 is A. 72 B. 120 C. 144 D. 5040 Updated April 2014 2. If a customer purchases 3 video games, 2 Sports games and 1 Classic game, the total number of way he can select the games is ______. Example 5 – Six Math 30-1 students (Abby, Brayden, Christine, Dallas, Ethan and Fran) are going to stand in a line. How many ways can they stand if: a) Fran must be in the third position? b) Brayden must be second and Ethan third? c) Dallas can’t be on either end of the line? d) Boys and girls alternate, with a boy starting the line? e) The first three positions are boys, the last three are girls? f) The row starts with two boys? g) The row starts with exactly two boys? h) Abby must be in the second position and a boy must be in the third? Updated April 2014 Example 6 -- The number of pathways, from A to C, passing through B, is ________. A. 6 B. 12 C. 18 D. 36 Updated April 2014 Example 7 -- If the only allowed directions are North and East, then the number of allowable paths from Point A to Point B, is A. 30 B. 60 C. 75 D. 90 Updated April 2014 Specific Outcome 3: Determine the number of combinations of n different elements take r at a time to solve problems. - Explain, using examples, the difference between a permutation and a combination - Determine the number of way that a subset of k elements can be selected from a set of n different elements - Determine the number of combinations of n different elements taken r at a time to solve problems - Explain why n must be greater than or equal to r in the notation nCr or ( ) - Explain, using examples, why nCr = nCn-r or ( ) ( ) - Be able to use both nCr and ( ) to solve problems A combination is a selection of objects in which the order of selection is not important, since the objects will not be arranged. ( ) ( ) When determining the number of possibilities in a situation, if order matters, it is a permutation. If order does not matter, it is a combination. Handshakes, committees, and teams playing each other are all combination problems. Updated April 2014 Example 1 -- How many different ways could 4 members be selected from a cheerleading squad with 12 members? Example 2 – There are 12 teams in a soccer league, and each team must play each other twice in a tournament. The number of games that will be played in total is: A. 6P2 B. 12C2 C. 12P2 D. 12C2 Example 3 – The number of committees consisting of 4 men and 5 women that can be formed from 10 men and 13 women is A. 10C4 13C5 B. 10P4 13P5 C. 22C9 D. 22P9 Example 4 – At a car dealership, the manager wants to line up 10 cars of the same model in the parking lot. There are 3 red cars, 2 blue cars, and 5 green cars. If all 10 cars are lined up in a row, facing forward, determine the number of possible car arrangements if the blue cars cannot be together. Example 5 – There are 12 people in line for a movie. If Shannon, Jeff and Chris are friends and will always stand together, the total number of possible arrangements for the entire line is A. B. C. 12P3 D. 12C3 Updated April 2014 Use the following information to answer the next question. ___________________________________________________________________________ If 14 different types of fruit are available, how many different fruit salads could be made using exactly 5 types of fruit? Student 1 Kevin used to solve the problem. Student 2 Ron suggested using 14P5. Student 3 Michelle solved the problem using 14C9. Student 4 Emma thought that 5P14 would give the correct answer. Student 5 John decided to use ( ). ___________________________________________________________________________ Example 6 -- The correct solution would be obtained by student number _____ and student number _____. Example 7 – If there are 36 handshakes in total, how many people were at the meeting? Updated April 2014 Example 8 -- How many different 4-letter arrangements are possible using any 2 letters from the word SPRING and any 2 letters from the word MATH? 1. If nPr = 6720 and nCr = 56, then the value of r is ____________. Updated April 2014 Specific Outcome 4: Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers). - Be able to show the relationship/patterns between the rows of Pascal’s triangle and the numerical coefficients of the terms in the expansion of a n binomial (x+y) - Explain how to determine the subsequent row in Pascal’s triangle, given any row - Relate the coefficients of the terms in the expansion of (x+y)n to the (n+1) row in Pascal’s triangle - Explain, using examples, how the coefficients of the terms in the expansion of (x+y)n are determined by combinations - Expand, using the binomial theorem, (x+y)n and determine a specific term in the expansion A combination is a selection of objects in which the order of selection is not important, since the objects will not be arranged.
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