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ALGEBRA 2: 10.1 Apply the Counting Principle and Permutations
Goal  Use the fundamental counting principle and find permutations.
VOCABULARY
Permutation
A permutation is an ordering of n objects.
Factorial
Represented by the symbol !, n factorial is defined as:
n! = n  {n  1)  {n  2) .....3  2  1.
FUNDAMENTAL COUNTING PRINCIPLE
Two Events If one event can occur in m ways and another event can occur in n ways, then the number of ways
that both events can occur is __m  n__ .
Three or More Events The fundamental counting principle can be extended to three or more events. For
example, if three events occur in m, n, and p ways, then the number of ways that all three events can occur is
__m  n  p__.
Example 1
Use the fundamental counting principle
Pizza You are buying a pizza. You have a choice of 3 crusts, 4 cheeses, 5 meat toppings, and 8 vegetable
toppings. How many different pizzas with one crust, one cheese, one meat, and one vegetable can you choose?
Checkpoint Complete the following exercise.
1. If the pizza crust was not a choice in Example 1, how many different pizzas could be made?
Example 2
Use the counting principle with repetition
Telephone Numbers A town has telephone numbers that all begin with 329 followed by four digits. How many
different phone numbers are possible (a) if numbers can be repeated and (b) if numbers cannot be repeated?
Example 3
The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters.
a.
How many different license plates are possible if letters and digits can be repeated?
b. How many different license plates are possible if letters and digits cannot be repeated?
Example 4
Find the number of permutations
Playoffs Eight teams are competing in a baseball playoff.
a. In how many different ways can the baseball teams finish the competition?
b. In how many different ways can 3 of the baseball teams finish first, second, and third?
PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME
The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr
n!
P 
n r ( n  r )!
Example 5
Find permutations of n objects taken rat a time
Homework You have 6 homework assignments to complete over the weekend. However, you only have time to
complete 4 of them on Saturday. In how many orders can you complete 4 of the assignments?
Checkpoint Complete the following exercises.
2. How many different 7 digit telephone numbers are possible if all of the digits can be repeated?
3. In Example 3, how many different ways can the teams finish if there are 6 teams competing in the
playoffs?
4. You were left a list of 9 chores to complete. In how many orders can you complete 5 of the chores?
PERMUTATIONS WITH REPETITION
The number of distinguishable permutations of n objects where one object is repeated s± times, another is
repeated s2 times, and so on, is:
n!
s1 !  s 2    s k !
Example 6
Find permutations with repetition
Find the number of distinguishable permutations of the letters in (a) EVEN and (b) CALIFORNIA.
Checkpoint Find the number of distinguishable permutations of the letters in the word.
5. TOMORROW
6. YESTERDAY
10.2 Use Combinations and the Binomial Theorem
Goal  Use combinations and the binomial theorem.
VOCABULARY
Combination
A selection of r objects from a group of n objects where the order is not important
Pascal's triangle
An arrangement of the values of nCr w a triangular pattern in which each row corresponds to a value of n
Binomial theorem
For any positive integer n, the binomial expansion of (a + b)n is
(a+ b)n = nC0anb0 + nC1an1b1 + nC2an2b2 +  + nCna0bn
COMBINATIONS OF n OBJECTS TAKEN r AT A TIME
The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr
nCr =
n!
(n  r )!  r!
Example 1
Find combinations
Books You are picking 7 books from a stack of 32. If the order of the books you choose is not important, how
many different 7 book groups are possible?
Example 2
A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit.
a. If the order in which the cards are dealt is not important,
how many different 5-card hands are possible?
b. In how many 5-card hands are all 5 cards of the same color?
Example 3
Decide to multiply or add combinations
Movie Rentals The local movie rental store is having a special on new releases. The new releases consist of 12
comedies, 8 action, 7 drama, 5 suspense, and 9 family movies.
a. You want exactly 2 comedies and 3 family movies. How many different movie combinations can you
rent?
b. You can afford at most 2 movies. How many movie combinations can you rent?
Example 4
William Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are
comedies, 10 are histories, and 10 are tragedies.
a. How many different sets of exactly 2 comedies and 1 tragedy can you read?
b. How many different sets of at most 3 plays can you read?
Checkpoint Complete the following exercises.
1. Find 7C4.
2. Find 6C3.
3. Find 12C11.
4. From Example 3, find the number of possible movie combinations if you can choose 2 action movies and
2 dramas.
Example 5
Solve a multi-step problem
Reading A popular magazine has 11 articles. You want to read at least 2 of the articles. How many different
combinations of articles can you read?
Example 6
During the school year, the girls’ basketball team is scheduled to play 12 home games. You want to attend
at least 3 of the games. How many different combinations of games can you attend?
Checkpoint Complete the following exercise.
5. Your school football team has 10 scheduled games for the season. You want to attend at least 4 games.
How many different combinations of games can you attend?
PASCAL'S TRIANGLE
The first and last numbers in each row are _1_. Every number other than _1_ is the sum of the closest two
numbers in the row directly above it.
Pascal's triangle:
As combinations
n = 0 (0th row)
0C _0_
n = 1 (1st row)
1C_0_ 1C_1_
n = 2 (2nd row)
2C_0_ 2C_1_ 2C_2_
n = 3 (3rd row)
3C_0_ 3C_1_ 3C_2_ 3C_3_
As numbers
_1_
_1_ _1_
_1_ _2_ _1_
_1_ _3_ _3_ _1_
Example 7
Use Pascal's triangle
Class Representatives Out of 5 finalists, your class must choose 3 class representatives. Use Pascal's triangle
to find the number of combinations of 3 students that can be chosen as representatives.
Checkpoint Complete the following exercise.
6. In Example 4, use Pascal's triangle to find the number of combinations of 3 students that can be chosen
from 8 finalists.
BINOMIAL THEOREM
 For
any positive integer n, the binomial expansion of (a + b)n is:
(a + b)n = nC0anb0 + nC1an  1b1 +    +nCna0bn
Notice that each term in the expansion of (a + b)n has the form nCran  1rbr where r is an integer from 0 to n.
Example 8
Expand a power of a binomial sum
Use the binomial theorem to write the binomial expansion. (x + 4)3
Example 9
Expand a power of a binomial difference
Use the binomial theorem to write the binomial expansion.
(2m  n)4
Example 10
Find a coefficient in an expansion
Find the coefficient of x5 in the expansion of (2x  7)9.
Checkpoint Use the binomial theorem to write the binomial expansion.
7. (a + 2b)3
8.
(6  s)4
9. Find the coefficient of x8 in the expansion of (3x  2)10.
10.3 Define and Use Probability
Goal  Find the likelihood that an event will occur.
VOCABULARY
Probability
A number from 0 to 1 that indicates the likelihood an event will occur
Theoretical probability
When all outcomes are equally likely, the number of outcomes in event A divided by the total number of
outcomes
Odds
Used to measure the chances in favor of an event occurring or the chances against an event occurring
Experimental probability
A calculation of the probability of an event based on performing an experiment
Geometric probability
Probability found by calculating a ratio of two lengths, areas, or volumes
THEORETICAL PROBABILITY OF AN EVENT
When all outcomes are equally likely, the theoretical probability that an event A will occur is:
Number of outcomes in event A
P(A)=
Total number of outcomes
The theoretical probability of an event is often simply called the probability of the event.
Example 1
You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number.
Checkpoint Complete the following exercises.
1. Using a standard deck of 52 playing cards, find the probability of drawing a red queen.
2. Using a standard deck of 52 playing cards, find the probability of a) picking an 8 and b) picking a red
king.
Example 2
You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the
given event.
a. A perfect square is chosen.
b. A factor of 30 is chosen.
ODDS IN FAVOR OF OR ODDS AGAINST AN EVENT
When all outcomes are equally likely, the odds in favor of an event A and the odds against an event A are:
Number of outcomes in A
Odds in favor of event A
=
Number of outcomes not in A
Number of outcomes not in A
Odds against event A
=
Number of outcomes in A
a
You can write the odds in favor of or against an event in b
the for or in the form a:b
Example 3
Find odds
Marbles A marble is drawn from a bag containing 6 red, 12 yellow, and 9 black marbles. Find (a) the odds in
favor of drawing a red marble and (b) the odds against drawing a black marble.
EXPERIMENTAL PROBABILITY OF AN EVENT
When an experiment is performed that consists of a certain number of trials, the experimental probability of an
event A is given by:
Number of trials where A occurs
P(A) =
Total number of trials
Example 4
Find an experimental probability
Exam Grades Exam grades of students in a history class are shown in the bar graph. Find the probability that a
randomly chosen student in this history class received a C or better.
1. A marble is drawn from a bag containing 6 blue, 6 yellow, 9 white, and 11 green marbles. Find the odds in
favor of drawing a yellow marble.
2. Using Example 4, find the experimental probability that a randomly selected student received a C or lower.
10.4 Find Probabilities of Disjoint and Overlapping Events
Goal  Find probabilities of compound events.
VOCABULARY
Compound Event
The union or intersection of two events
Disjoint or mutually exclusive events
Two events that have no outcomes in common
PROBABILITY OF COMPOUND EVENTS
If A and B are two events, then the probability of A or B is:
P(A or B) = _P(A) + P(B)  P(A and B)_
If A and B are disjoint events, then the probability of A or B is:
P(A or B) = _P(A) + P(B)_
Example 1
Find probability of disjoint events
You roll a six-sided number cube. What is the probability of rolling a 2 or a 5?
Checkpoint Complete the following exercise.
1. You roll a six-sided number cube. What is the probability of rolling a 1 or an even number?
Example 2
Find probability of compound events
You roll a six-sided number cube. What is the probability of rolling an odd number or a number less
than 3?
Example 3
Use a formula to find P(A and B)
Music In a survey of 300 students, 150 like pop music or country music. There are 97 students who like pop
music and there are 83 students who like country music. What is the probability that a randomly selected
student likes both pop music and country music.
Example 4
Find the probability of complements
When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that the sum is
not 4 and the sum is greater than or equal to 3.
PROBABILITY OF THE COMPLEMENT OF AN EVENT
The probability of the complement of A is P( A ) = _1  P(A)_.
Checkpoint Complete the following exercises.
2. You roll a six-sided number cube. What is the probability of rolling a number less than 4 or an even
number?
3. In a survey of 125 people, 90 of them like orange juice or grape juice. There are 62 people who like
orange juice and 43 people who like grape juice. What is the probability that a randomly selected person
likes both orange juice and grape juice?
4. From Example 4, find the probability that the sum is not 8.
10.5 Find Probabilities of Independent and Dependent Events
Goals  Examine independent and dependent events.
VOCABULARY
Independent events
Two events such that the occurrence of one has no effect on the occurrence of the other
Dependent events
Two events such that the occurrence of one affects the occurrence of the other
Conditional probability
The probability that event B will occur given that event A has occurred is called the conditional probability of B
given A.
PROBABILITY OF INDEPENDENT EVENTS
If A and B are independent events, then the probability that both A and B occur is:
P(A and B) = _P(A)  P(B)_
More generally, the probability that n independent events occur is the _product_ of the n probabilities of the
individual events
Example 1
Find probability of three independent events
Attendance Every morning, one student in a class of 24 students is randomly chosen to take attendance. What
is the probability that the same student will be chosen three days in a row?
Checkpoint Complete the following exercises.
1. During a high school track meet, each race consists of 9 competitors who are randomly assigned lanes
from 1 to 9. What is the probability that a runner will draw lanes 1, 2, or 3 in the three races in which he
competes?
PROBABILITY OF DEPENDENT EVENTS
If A and B are dependent events, then the probability that both A and B occur is:
P(A and B) = __P(A)  P(B  A)__
Example 3
Find a conditional probability
Eye Color Find the probability that (a) a listed person has blue eyes and (b) a male has blue eyes.
Green eyes
Blue eyes
Brown eyes
Hazel eyes
Male
27
35
15
23
Female
12
9
38
41
Example 4
Comparing independent and dependent events
You randomly select two marbles from a bag containing 15 yellow, 10 red, and 12 blue marbles. What is
the probability that the first marble is yellow and the second marble is not yellow if (a) you replace the
first marble before selecting the second, and (b) you do not replace the first marble?
Example 5
Find probability of three dependent events
Pencils Your teacher passes around a box with 10 red pencils, 8 pink pencils, and 13 green pencils. If you and
the two people in your group are the first to randomly select a pencil, what is the probability that all three of you
select pink pencils?
Checkpoint Complete the following exercises.
2. Use the table in Example 3 to find the probability that a female has hazel eyes.
3. From Example 4, find the probability that both marbles will be yellow if (a) you replace the first marble
and (b) you do not replace the first marble.
4. From Example 5, what is the probability that you and your group members all choose a red pencil?
11.1 Find Measures of Central Tendency and Dispersion
Goal  Describe data using statistical measures.
VOCABULARY
Statistics
Numerical values used to summarize and compare sets of data
Measure of central tendency
A number used to represent the center or middle of a set of data values. This is represented by the mean,
median, and mode.
Measure of dispersion
A statistic that tells you how dispersed, or spread out, data values are
Standard deviation
A measure that describes the typical difference (or deviation) between a data value and the mean
Outlier
A value that is much greater than or much less than most of the other values in a data set
MEASURES OF CENTRAL TENDENCY

The mean, or __average__ , of n numbers is the __sum__ of the numbers __divided__ by n. The mean is
denoted by x , which is read as "x-bar." For the data set x1,x2,…xn, the mean is
x
x 1  x 2  ...  xn
n
.

The median of n numbers is the __middle__ number when the numbers are written in order. (If n is even, the
median is the __mean__ of the two middle numbers.)

The mode of n numbers is the number or numbers that occur __most frequently__. There may be _one_
mode, __no__ mode, or __more than one__ mode.
Example 1
Find measures of central tendency
Quiz Scores The data sets at the right give quiz scores for two different biology classes. Find the mean, median,
and mode of each data set.
Class A
Class B
15, 17, 17,
17, 18, 19,
21, 22, 25
16, 18, 19,
21, 22, 22,
22, 24, 25
STANDARD DEVIATION OF A DATA SET
The standard deviation  (read as "sigma") of x1, x2,…xn is:
=
x
1
 
2

2

 x  x 2  x  ...  xn  x

2
n
Example 2
Find the range and standard deviation
Find the range and standard deviation for the quiz scores in each data set from Example1.
Example 3
Examine the effect of an outlier
Soccer The winning scores for the first 9 games of the soccer season are: 3, 4, 2, 5, 3,1, 4, 3, 2.
a. Find the mean, median, mode, range, and standard deviation of the data set.
b. The winning score in the next game is an outlier, 9. Find the new mean, median, mode, range, and standard
deviation.
c. Which measure of central tendency does the outlier affect the most? the least?
d. What effect does the outlier have on the range and standard deviation?
Checkpoint Complete the following exercises.
The data set below gives the recorded speeds (in mi/h) for 10 different cars on a local highway.
69, 62, 64, 67, 62, 64, 63, 65, 60, 64
1. Find the range and standard deviation of the data set.
2. Find the range and standard deviation of the data set
3. The next car that drives by is having car trouble, so the recorded speed is 36 mi/h. Find the new mean,
median, mode, range, and standard deviation.