Updated Jan 2016

SESSION 4

Permutations Combinations

Polynomials

Updated Jan 2016

Updated Jan 2016

Mathematics 30-1 Learning Outcomes

Permutations and Combinations

General Outcome: Develop algebraic and numeric reasoning

that involves combinatorics.

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 𝑎 × 𝑏 ×

𝑐 × …. 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.

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.

Updated Jan 2016

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. 𝟐! × 𝟓! × 𝟐!

B. 𝟐! × 𝟓!

C. 7P2 × 5P5

D. 2! × 6!

Updated Jan 2016

Factorial notation is an abbreviation for products of successive positive

integers.

𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) … × 3 × 2 × 1, 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ 𝑁 𝑎𝑛𝑑 0! = 1 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.

𝑛𝑃𝑟 = 𝑛!

(𝑛−𝑟)!

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 𝑛!

𝑎!𝑏!𝑐!… 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.

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

Updated Jan 2016

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 digits 2, 3, 5, 6 and 7 with no digits repeating is

A. 72

B. 120

C. 144

D. 5040

Updated Jan 2016

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 Jan 2016

Example 6 -- The number of pathways, from A to C, passing through B, is ________.

A. 6

B. 12

C. 18

D. 36

Updated Jan 2016

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 Jan 2016

A combination is a selection of objects in which the order of selection is not

important, since the objects will not be arranged.

𝑛𝐶𝑟 = 𝑛!

(𝑛−𝑟)!𝑟! nCr = (𝑛

𝑟) ; 𝑛 ≥ 𝑟, 𝑟 ≥ 0

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.

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 (𝑛𝑟

)

- Explain, using examples, why nCr = nCn-r or (𝑛𝑟

) = ( 𝑛𝑛−𝑟

)

- Be able to use both nCr and (𝑛𝑟

) to solve problems

Updated Jan 2016

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 × 2

B. 12C2

C. 12P2 × 2

D. 12C2 × 2

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. 3! × 2! × 9!

B. 3! × 10!

C. 12P3 × 9!

D. 12C3 × 9!

Updated Jan 2016

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 14! 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 (145

).

___________________________________________________________________________

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 Jan 2016

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 Jan 2016

A combination is a selection of objects in which the order of selection is not

important, since the objects will not be arranged.

In the expansion of (x+y)n, the general term is 𝑡𝑘+1 =nCk (x)n-k

(y)k

In the expansion of (x+y)n, where 𝑛 ∈ 𝑁, the coefficients are identical to

the numbers in the (𝑛 + 1)𝑡ℎ row of Pascal’s triangle.

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

binomial (x+y)n , 𝑛 ∈ 𝑁

- 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

Updated Jan 2016

Example 1 – Find the value of a if the expansion of (4𝑥 − 7)(3𝑎+2) has 21 terms.

______________________________________________________________________

Example 4 – The three statements that are true are numbered ____, ___ and ____.

Updated Jan 2016

Example 5 -- Which of the following represents the 3rd term in the expansion

(2 − 3𝑥)7?

A. (72)(2)2(−3𝑥)5

B. (73)(2)3(−3𝑥)4

C. (73)(2)4(−3𝑥)3

D. (72)(2)5(−3𝑥)2

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 (3𝑎 + 𝑏2)10, what is the coefficient of the term

containing 𝑎4𝑏12?

Updated Jan 2016

Example 8 -- What is the coefficient of the term containing 𝑥2 in the expansion of

(𝑥 + 3)7?

Example 9 – Given that a term in the expansion of (𝑎𝑥 − 𝑦)6 is −252𝑥𝑦5, determine

the numerical value of a.

1. The expansion of (3𝑥2 − 2𝑦3)3𝑘−9 has 22 terms. The value of k is, to the

nearest whole number, ___________.

Updated Jan 2016

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 _________.

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.

Updated Jan 2016

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.

Updated Jan 2016

Key Concepts

A polynomial has the form 1 2

1 2 1 0...x x

n nf x a x a x a x a x a

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: x

Range: y

Degree Leading Coefficient

Constant (y-intercept) 3 23 5 4 7y x x x

Updated Jan 2016

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: x

Range: depends on max. value

Updated Jan 2016

+

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.

On way of expressing the Polynomial is in the form:

P x R

Q xx a x a

or

( )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.

Updated Jan 2016

Multiplicity – the multiplicity of a zero or x-intercept, corresponds to the number of times a factor is

repeated

Updated Jan 2016

Updated Jan 2016

Examples:

1. For each polynomial function, state the degree. If the function is not a polynomial, indicate why

a) 1

5h xx

b) 24 3 8y x x c) 69g x x d) 3f x x

2. Complete the table

Function Leading

Coefficient Degree

Odd or

Even

End Behaviour

Possible Number of x-int.

y-int.

Constant Term

Max. or Min. or Neither

3 28 7 1x x x

4 2 10x x x

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 3 24 6x x x is divided by the

binomial 1x . Check using synthetic division.

Updated Jan 2016

3. You can model the volume, in cubic centimetres, of a rectangular box by the polynomial function

3 23 12 4V x x x x . Determine expressions for the other dimensions of the box if the

height is 2.x

4. Using the Remainder Theorem, find the value of k in the polynomial 3 25 8x x kx is divided by

3x the remainder is 1. Determine the value of k.

5. If 3 2 6P x x ax bx with 4 6P and 2 0P , find the values of a and b. SE

6. Use the Remainder Theorem to determine the remainder when 3 3 10x x is divided by each polynomial:

a) 2x b) 5x

Updated Jan 2016

7. Which of the following are factors of the polynomial 3 24 6?P x x x x

a) 1x b) 2x

8. Algebraically factor the following fully:

a) 3 22 3 3 2x x x b) 3 2 8 12x x x

9. The back of a van has a volume V(w) that can be represented by the expression

3 28 20 16V w w w w , 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?

Updated Jan 2016

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 (-)

Updated Jan 2016

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.

Updated Jan 2016

Practice Test

1. Which statement is true of 3 23 4 2 9P x x x x ?

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. 23 3 5 17 42P x x x x

2. Which set of values for x should be tested to determine the possible zeros of 4 3 22 7 8 12x x x x ?

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 3 22 5 9 18x x x ?

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?

A. 2

3 1 2x x x

B. 3 1 2x x x

C. 3 1 2x x x

D. 2

3 1 2x x x

5. The graph of the function 4 2 6f x x 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

Updated Jan 2016

6. If the zeros of a polynomial are -1, 1

2 and

2

3, then the polynomial could be

A. 3 212 2 10 4x x x

B. 3 26 5 2x x x

C. 3 218 3 15 6x x x

D. 3 230 5 25 10x x x

7. Which of the following is a factor of 4 3 24 8 40f x x x x ?

A. (x + 2) B. (x – 4) C. (x – 6) D. (x + 8)

8. If 4

03

P

and 2 0P , then a second degree factor of P(x) is:

A. 23 2 8x x

B. 23 2 8x x

C. 24 5 6x x

D. 24 11 6x x

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. 2 1 2f x x x x

B. 2 1 2f x x x

C. 2 1 2f x x x x

D. 2 1 2f x x x x

Updated Jan 2016

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

B. 2

p x x a x b x c

C. 2

p x x a x b x c

D. 2

p x x 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

A. 3

4

B. 1

2

C. – 4 D. – 10

12. The graph of the function 3 22y x bx cx d could be:

A. B.

C. D.

Updated Jan 2016

Use the following information to answer the next two questions.

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 3 23 4,f x x x kx the value of k is .

The graph of the polynomial function y = f(x) is shown.

Updated Jan 2016

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

Updated Jan 2016

Use the following functions to answer the next question.

1 4 310 2 5y x x x

2 3 2 13 2 4y x x x

3 5 3y x

4 3 2 14 2y x x

x

5

5 4 32 7 3 2 7xy x x x

6. Which of the functions above represent polynomial functions? ______

Use the following information to answer the next two questions.

7. If the polynomial function is written in the form 2

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

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.

Updated Jan 2016

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 2 3⁄ ,

given that f(2) = -112.

Updated Jan 2016

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) 0 5x

c) 2.11

2. 4 3 5 3 2y x x x