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.
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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
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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.
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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.
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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
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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?
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Example 6 -- The number of pathways, from A to C, passing through B, is ______.
A. 6 B. 12 C. 18 D. 36
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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
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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.
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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
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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?
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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 ______.
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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. n n-k k In the expansion of (x+y) , the general term is nCk (x) (y)
In the expansion of (x+y)n, where , the coefficients are identical to the numbers in the ( ) row of Pascal’s triangle.
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Example 1 – Find the value of a if the expansion of ( )( ) has 21 terms.
Example 2 -- Which of the following represents the 3rd term in the expansion ( ) ?
A. ( )( ) ( )
B. ( )( ) ( )
C. ( )( ) ( )
D. ( )( ) ( )
Example 3 – A child who is going on a trip is told that out of his 8 favorite toys, he can bring at most three toys. The number of ways he could select which toys he brings is
A. 8P0 + 8P1 + 8P2 + 8P3 B. 8C0 + 8C1 + 8C2 + 8C3 C. 8C3 - (8C0 + 8C1 + 8C2) D. 8C0 8C1 8C2 8C3
______
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Example 4 – The three statements that are true are numbered ____, ___ and ____.
Example 5 -- Which of the following represents the 3rd term in the expansion ( ) ?
A. ( )( ) ( )
B. ( )( ) ( )
C. ( )( ) ( )
D. ( )( ) ( )
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Example 6 – A child who is going on a trip is told that out of his 8 favorite toys, he can bring at most three toys. The number of ways he could select which toys he brings is
A. 8P0 + 8P1 + 8P2 + 8P3 B. 8C0 + 8C1 + 8C2 + 8C3 C. 8C3 - (8C0 + 8C1 + 8C2) D. 8C0 8C1 8C2 8C3
Example 7 – In the expansion of ( ) , what is the coefficient of the term containing ?
Example 8 -- What is the coefficient of the term containing in the expansion of ( ) ?
Example 9 – Given that a term in the expansion of ( ) is , determine the numerical value of a.
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1. The expansion of ( ) has 22 terms. The value of k is, to the nearest whole number, ______.
2. The number of ways 3 tiles can be pulled out of a bag containing 20 tiles, is the same as the number of ways k tiles can be pulled out of 20 tiles. The value of k is ______.
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Polynomial Functions
RF12. Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5 ).
12.1 Identify the polynomial functions in a set of functions, and explain the reasoning.
12.2 Explain the role of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function.
12.3 Generalize rules for graphing polynomial functions of odd or even degree.
12.4 Explain the relationship between:
- the zeros of a polynomial function
- the roots of the corresponding polynomial equation
- the x-intercepts of the graph of the polynomial function.
12.5 Explain how the multiplicity of a zero of a polynomial function affects the graph.
12.6 Sketch, with or without technology, the graph of a polynomial function.
12.7 Solve a problem by modelling a given situation with a polynomial function and analyzing the graph of the function.
Notes:
- Students must understand the relationship between zeros of a function, roots of an equation, x- intercepts of a graph, and factors of a polynomial.
- Analyzing a polynomial function graphically includes: leading coefficients, maximum and minimum points, domain, range, x- and y-intercepts, zeros, multiplicity, odd and even degrees, and end behaviour.
- Students should be able to identify when no real roots exist, but the calculation of them is beyond the scope of this outcome.
- The term “maximum and minimum point” refers to the absolute maximum and absolute minimum point.
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Key Concepts
xx12 A polynomial has the form fx axnn axax12 ... 1 axa 0 Degree Leading Coefficient yxxx 332 54 7 Constant (y-intercept)
Degree: Odd (1, 3, or 5, …) Characteristics
(+) Leading Coefficient
Graph extends from quadrant III to I No Absolute max. or min. points Number of possible x-intercepts is from 1 to degree Domain: x Range: y
Degree: Odd (1, 3, or 5, …) Characteristics
(–) leading coefficient
Graph extends from quadrant II to IV No absolute max. or min. points Number of possible x-intercepts is from 1 to degree Domain: Range:
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Degree: Even (2, or 4, …) Characteristics
(+) Leading Coefficient
Graph extends from quadrant II to I Contains an absolute minimum Number of possible x-intercepts is from 0 to degree Domain: x Range: depends on min. value
Degree: Even (2, or 4, …) Characteristics
(–) Leading Coefficient
Graph extends from quadrant III to IV Contains an absolute maximum Number of possible x-intercepts is from 0 to degree Domain: Range: depends on max. value
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The Remainder Theorem
When a polynomial P(x) is divided by x – a, and the remainder is a constant, the remainder is equal to P(a). Because of this theorem, we can find the remainder of a polynomial without having to work through long division or synthetic division.
Px R On way of expressing the Polynomial is in the form: Qx or x ax a P x () x a Q x R
The Factor Theorem
The factor theorem states that x – a is a factor of a polynomial P(x) if and only if P(a) = 0. If and only if means that the result works both ways. That is, if x – a is a factor, then P(a) = 0 and if P(a) = 0, then x – a is a factor of a polynomial P(x).
Integral Zero Theorem
If a polynomial P(x) has any factor of the form (x – k), then k is a factor of the constant term of the polynomial. This means we can look to factor of the constant term as the values of k to test. The values of k are called integral zeros.
When factoring higher degree polynomials, the use of only the integral zero theorem can be time consuming. Always check if grouping is possible. Once one factor is found, use synthetic or long division to find others.
Equations and Graphs of Polynomial Functions +
The zeros of any polynomial function correspond to the x-intercept of the graph and to the root of the corresponding equation.
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Multiplicity – the multiplicity of a zero or x-intercept, corresponds to the number of times a factor is repeated
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Examples:
1. For each polynomial function, state the degree. If the function is not a polynomial, indicate why 1 a) hx 5 b) yxx4382 c) g xx 9 6 d) f xx 3 x
2. Complete the table Odd Possible Max. or Leading End y- Constant Function Degree or Number Min. or Coefficient Behaviour int. Term Even of x-int. Neither
xxx32 87 1
x42 x x 10
Examples:
1. What is the corresponding binomial factor of a polynomial, P(x), given the value of the zero? a) P(2) = 0 b) P(-4) = 0
2. Use the Remainder Theorem to determine the remainder when x3246 x x is divided by the binomial x 1. Check using synthetic division.
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3. You can model the volume, in cubic centimetres, of a rectangular box by the polynomial function V x 312 x32 xx 4 . Determine expressions for the other dimensions of the box if the height is x 2.
4. Using the Remainder Theorem, find the value of k in the polynomial xxkx3258is divided by x 3 the remainder is 1. Determine the value of k.
5. If P x x32 ax bx 6 with P46 and P20 , find the values of a and b. SE
6. Use the Remainder Theorem to determine the remainder when xx3 3 10 is divided by each polynomial: a) x 2 b) x 5
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7. Which of the following are factors of the polynomial P x xx32 46? x a) x 1 b) x 2
8. Algebraically factor the following fully: a) 2332xxx32 b) xxx32 812
9. The back of a van has a volume V(w) that can be represented by the expression V w w32 8 w 20 w 16 , where V is the volume and w is the width of the back end of the van. a) What are the factors that represent the possible dimensions, in terms of w, of the van?
b) If w = 2, what are the dimensions of the cube van?
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Examples:
1. Identify the specific properties of the following graphs.
Degree:
Factors:
Multiplicity
multiplicity
End Behaviour:
Leading Coefficent:
Interval where function is (+)
Interval where function is (-)
Degree:
Factors:
Multiplicity
Multiplicity
End Behaviour:
Leading Coefficent:
Interval where function is (+)
Interval where function is (-)
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2. Determine the equation with the least degree for each polynomial function. a) quartic function with zeros 2 (multiplicity 3) and -5, and y-intercept 30
b) quintic function with zeros -1 (multiplicity 2), 3 (multiplicity 1), and -2 (multiplicity 2), and a constant term -12.
3. Sketch the graph of a fifth degree polynomial function with one real root of multiplicity of 3 and with a negative leading coeffcient.
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Practice Test 1. Which statement is true of P x 3 x32 4 x 2 x 9 ?
A. When P(x) is divided by x + 1, the remainder is 6 B. x – 1 is a factor of P(x) C. P(3) = 36 D. P x xx 3 3 x 2 5 17 42
2. Which set of values for x should be tested to determine the possible zeros of xxxx4322 78 12 ?
A. 1, 2, 4, 12 B. 1, 2, 3, 4, 12 C. 1, 2, 3, 4, 6, 8 D. 1, 2, 3, 4, 6, 12
3. Which of the following is a factor of 2x32 5 x 9 x 18 ?
A. x – 1 B. x + 2 C. x + 3 D. x – 6
4. Which polynomial function has zeros of 3, 1, and 2, and y-intercept of y = -6?
2 A. x3 x 1 x 2 B. x3 x 1 x 2 C. x3 x 1 x 2 2 D. x3 x 1 x 2
5. The graph of the function f x x 426 x x is transformed by a horizontal stretch by a factor of two. Which of these statements are true?
A. The new zeros of the function are -12, -8, 4. B. The new zeros of the function are -3, -2, 1. C. The new y-intercept is -96 D. The new y-intercept is -24
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1 2 6. If the zeros of a polynomial are -1, and , then the polynomial could be 2 3 A. 12210xxx32 4 B. 6x32 x 5 x 2 C. 18315xxx32 6 D. 30525xxx32 10
7. Which of the following is a factor of f x 48 x4 x 403 x 2 ?
A. (x + 2) B. (x – 4) C. (x – 6) D. (x + 8)
4 8. If P 0 and P20 , then a second degree factor of P(x) is: 3 A. 328xx2 B. 3xx2 2 8 C. 4xx2 5 6 D. 4xx2 11 6
9. A student used the graph of a third degree polynomial function to make the table of values below. x -3 -2 -1 0 1 2 3 f(x) -24 0 4 0 0 16 60
The equation for this function is: A. f x 2 x x 1 x 2 B. f x 2 x 1 x 2 C. f x 2 x x 1 x 2 D. f x 2 x x 1 x 2
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10. The equation of the polynomial function shown below, assuming a, b, c are positive integers, could be:
A. p x x a x b x c 2 B. p x x a x b x c 2 C. p x x a x b x c 2 D. p xx a x b x c
11. The graph of a polynomial function of the form P x a x s x q x r has x-intercept of -1, -2, and 4. If the y-intercept is 32, then the value of a is
3 A. 4 1 B. 2 C. – 4 D. – 10
12. The graph of the function y2 x32 bx cx d could be:
A. B.
C. D.
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Use the following information to answer the next two questions.
The graph of the polynomial function y = f(x) is shown.
Numerical Response:
1. What is the minimum possible degree for the polynomial function above is .
2. When the above function is written in factored form it is expressed as 2 f x a x b x c x d , where a, b, c, and d are all positive. The value of a to the nearest tenth is .
3. If x + 2 is a factor of f x x32 3 x kx 4, the value of k is .
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4. The graph of a polynomial function has 3 distinct negative real roots and 2 equal positive real roots. The minimum degree of this function is .
5. Match three of the graphs numbered above with a statement below that best describes that function
The graph that has a positive leading coefficient is graph number _
The graph of the function that has two different zeros, each with a multiplicity of 2 is graph number _____
The graph that could be a degree of 4 function is graph number
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Use the following functions to answer the next question.
1 y xxx43 102 5
2 y324 xx321 x
3 yx53
1 4 yxx4232 x
5
y 2 x5 7 x 4 3 x 3 2x 7
6. Which of the functions above represent polynomial functions? ______
Use the following information to answer the next two questions.
The partial graph of a fourth degree polynomial
function P(x) is shown. The leading coefficient
is 1 and the x-intercepts of the graph are integers.
2 7. If the polynomial function is written in the form P x a x b x c x d where a, b, c, and d are all positive integers, then the values of a, b, c, and d are .
8. The graph has a y-intercept of -m, the value of m is
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Written Response
1. A box with no lid is made by cutting four squares of side length x from each corner of a 10 cm by 20 cm rectangular sheet of metal. a) Find and expression that represents the volume of the box.
b) Sketch a graph and state the restrictions.
c) Find the value of x, to the nearest hundredth of a centimetre, that gives the maximum volume.
2. Determine the equation of the cubic function, in factored form, whose roots are 3, -5, and ⁄ , given that f(2) = -112.
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Polynomials Practice Test Answers:
MULTIPLE CHOICE:
1. B 2. D 3. B 4. B 5. A 6. D 7. A 8. A 9. C 10. C 11. C 12. D NUMERICAL RESPONSE:
1. 4 2. 0.1 3. 4 4. 5 5. 423 6. 13 7. 1143 8. 12 WRITTEN RESPONSE:
1. a) V=x(10-2x)(20-2x) b) 05x
c) 2.11
2. y4 x 3 x 5 3 x 2
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