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
D.
5P5
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.
B.
C.
D.
{2}
{3}
{2, 5}
{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.
B.
C.
D.
72
120
144
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.
B.
C.
D.
6
12
18
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.
B.
C.
D.
30
60
75
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.
B.
C.
D.
6P2
12C2
12P2
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.
B.
C.
D.
10C4
13C5
10P4
13P5
22C9
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.
D.
12P3
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
binomial (x+y)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
A combination is a selection of objects in which the order of selection is not
important, since the objects will not be arranged.
n-k
k
 In the expansion of (x+y)n, the general term is
nCk (x) (y)

In the expansion of (x+y)n, where
, the coefficients are identical to
) row of Pascal’s triangle.
the numbers in the (
Updated April 2014
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.
B.
C.
D.
8P0
+ 8P1 + 8P2 + 8P3
8C0 + 8C1 + 8C2 + 8C3
8C3 - (8C0 + 8C1 + 8C2)
8C0
8C1
8C2
8C3
______________________________________________________________________
Updated April 2014
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. ( )( ) (
Updated April 2014
)
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.
B.
C.
D.
8P0
+ 8P1 + 8P2 + 8P3
C
8 0 + 8C1 + 8C2 + 8C3
8C3 - (8C0 + 8C1 + 8C2)
8C0
8C1
8C2
8C3
Example 7 – In the expansion of (
containing
?
) , what is the coefficient of the term
Example 8 -- What is the coefficient of the term containing
(
) ?
Example 9 – Given that a term in the expansion of (
the numerical value of a.
Updated April 2014
in the expansion of
) is
, determine
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 _________.
Updated April 2014
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, xintercepts 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 April 2014
Key Concepts
 A polynomial has the form f  x   an x x  an1 x x 1  ...  a2 x 2  a1 x  a0
Degree
Leading Coefficient
y  3x3  5x2  4 x  7
Degree: Odd (1, 3, or 5, …)
Constant (y-intercept)
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





Updated April 2014
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:
Even (2, or 4, …)
Characteristics
(+) Leading Coefficient





Degree:
Even (2, or 4, …)
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
Characteristics
(–) Leading Coefficient





Updated April 2014
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
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   ( x  a)Q  x   R
P  x
R
 Q  x 
or
xa
xa
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 April 2014
Multiplicity – the multiplicity of a zero or x-intercept, corresponds to the number of times a factor is
repeated
Updated April 2014
Updated April 2014
Examples:
1. For each polynomial function, state the degree. If the function is not a polynomial, indicate why
a)
h  x5 
1
x
b) y  4 x  3x  8
2
c) g  x   9 x 6
d) f  x  
3
x
2. Complete the table
Function
Leading
Coefficient
Degree
Odd
or
Even
End
Behaviour
Possible
Number
of x-int.
yint.
Constant
Term
Max. or
Min. or
Neither
 x3  8 x 2  7 x  1
x4  x2  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 x3  4 x2  x  6 is divided by the
binomial x  1 . Check using synthetic division.
Updated April 2014
3. You can model the volume, in cubic centimetres, of a rectangular box by the polynomial function
V  x   3x3  x 2  12 x  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 x3  5x2  kx  8 is divided by
x  3 the remainder is 1. Determine the value of k.
5. If P  x   x3  ax 2  bx  6 with P  4   6 and P  2   0 , find the values of a and b. SE
6. Use the Remainder Theorem to determine the remainder when x3  3x  10 is divided by each
polynomial:
a) x  2
b) x  5
Updated April 2014
7. Which of the following are factors of the polynomial P  x   x3  4 x 2  x  6?
a)
x 1
8. Algebraically factor the following fully:
a) 2 x3  3x2  3x  2
b) x  2
b) x3  x2  8x  12
9. The back of a van has a volume V(w) that can be represented by the expression
V  w  w3  8w2  20w  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?
Updated April 2014
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 April 2014
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 April 2014
Practice Test
1. Which statement is true of P  x   3x3  4 x 2  2 x  9 ?
A.
B.
C.
D.
When P(x) is divided by x + 1, the remainder is 6
x – 1 is a factor of P(x)
P(3) = 36
P  x    x  3  3x 2  5x  17   42
2. Which set of values for x should be tested to determine the possible zeros of
x4  2 x3  7 x2  8x  12 ?
A.
B.
C.
D.
1, 2, 4, 12
1, 2, 3, 4, 12
1, 2, 3, 4, 6, 8
1, 2, 3, 4, 6, 12
3. Which of the following is a factor of 2 x3  5x2  9 x  18 ?
A.
B.
C.
D.
x–1
x+2
x+3
x–6
4. Which polynomial function has zeros of 3, 1, and 2, and y-intercept of y = -6?
A.
B.
C.
D.
 x  3 x  1  x  2
 x  3 x  1 x  2
 x  3 x  1 x  2
2
 x  3 x  1 x  2
2
5. The graph of the function f  x    x  4  x  2  x  6  is transformed by a horizontal stretch
by a factor of two. Which of these statements are true?
A.
B.
C.
D.
The new zeros of the function are -12, -8, 4.
The new zeros of the function are -3, -2, 1.
The new y-intercept is -96
The new y-intercept is -24
Updated April 2014
6. If the zeros of a polynomial are -1,
A.
B.
C.
D.
1
2
and , then the polynomial could be
2
3
12 x3  2 x2  10 x  4
6 x3  x 2  5 x  2
18x3  3x2  15x  6
30 x3  5x2  25x  10
7. Which of the following is a factor of f  x   4 x 4  x3  8x 2  40 ?
A.
B.
C.
D.
(x + 2)
(x – 4)
(x – 6)
(x + 8)
 4 
  0 and P  2   0 , then a second degree factor of P(x) is:
 3 
A. 3x 2  2 x  8
B. 3x 2  2 x  8
C. 4 x 2  5x  6
D. 4 x2  11x  6
8. If P 
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
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 
Updated April 2014
4
0
0
16
60
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.
p  x    x  a   x  b  x  c 
C.
p  x     x  a   x  b  x  c 
D.
p  x     x  a   x  b  x  c 
2
2
2
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.
B.
3
4
1
2
C. – 4
D. – 10
12. The graph of the function y  2 x  bx  cx  d could be:
3
2
A.
B.
C.
D.
Updated April 2014
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
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
2
tenth is
.
3. If x + 2 is a factor of f  x   x3  3x 2  kx  4, the value of k is
Updated April 2014
.
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 April 2014
Use the following functions to answer the next question.
1
y  x4  10 x3  2 x  5
2
y  3x3  2 x2  x1  4
3
y  5x  3
4
y  4 x3  2 x 2 
1
x
5
y  2 x5  7 x4  3x3  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.
7. If the polynomial function is written in the form P  x   a  x  b 
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
Updated April 2014
2
 x  c  x  d  where a, b, c,
.
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.
Updated April 2014
Polynomials Practice Test Answers:
MULTIPLE CHOICE:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
B
D
B
B
A
D
A
A
C
C
C
D
1.
2.
3.
4.
5.
6.
7.
8.
4
0.1
4
5
423
13
1143
12
NUMERICAL RESPONSE:
WRITTEN RESPONSE:
1. a) V=x(10-2x)(20-2x)
b) 0  x  5
c) 2.11
2.
y  4  x  3 x  5 3x  2 
Updated April 2014