Notes key

Algebra 2: Chapter 10 Notes Packet
Name:______________________________
**Included inside: Assignment Sheet, Notes, Study Guide, Homework Worksheets**
Algebra 2 Chapter 10 Assignment Sheet
*All assignments are due the day after they are assigned*
Date
26
1/3
M
In-Class
10.1: The Counting Principle and
Permutations
Homework
pg. 686 #10-28even, 30-39 x3, 44, 46, 50
Wkst #1 all
27
1/4
T
10.2: Combinations and Binomial
Theorem
pg. 687: 7-17odd
pg. 694 #2-10even, 13-18all, 25-31odd, 32-34all
Ch. 10 Worksheet #2
28
1/5
W
10.3: Probability
pg. 701 #3-16all
Ch. 10 Worksheet #3
29
1/6
Th
10.4: Disjoint and Overlapping Events
pg. 710 #20-25all, 35-38all
Ch. 10 Worksheet #4
30
1/7
F
10.5: Independent and Dependent Events
pg. 721#9-14all, 26-30 even
Ch. 10 Worksheet #5
Chapter 10 Test on Tuesday
31
1/10
M
Review
Chapter 10 Study Guide
32
1/11
T
Chapter 10 Test
Print Ch. 12 Notes by Wednesday
Chapter 10 Test on Tuesday
Ch. 10 Cumulative Review #3, 5-6, 9, 15, 17, 18,
20-24, 26-28, 33, 35, 38, 40-41, 44-46
Print Ch. 12 Notes by Wednesday
1
Chapter 10 Notes Packet
Name:______________________________
Notes #26— Applying the Counting Principle and Permutations (Section 10.1)
Ex 1: (a) Katie, Mark, and Nate still need to present their individual projects in Biology class. Make a
list of all possible orders in which they can present. In how many different orders can they present
their project?
(b) If Katie needs to present last, in how many different orders can they present their project?
Ex 2: You are going out to eat pizza for your birthday. You have three choices to make: (1) crust:
regular or thin crust, (2) cheese: regular, low fat, or extra cheese, and (3) toppings: vegetarian, BBQ
chicken, or pepperoni. Make a tree diagram to illustrate all of your pizza options. (a) In how many
different ways can you choose your pizza?
Crust
Cheese
Toppings
(b) Using the fundamental counting principle below, find the number of different ways you can choose
your pizza if they are out of pepperoni.
Fundamental Counting Principle: (Use this principle when you are finding the number of
choices available or the number of orders possible.)
Two Events: If one event can occur in m ways and another event can occur in n ways, then both
events can occur in m  n ways.
Three or More Events: If three events can occur in m, n, and p ways, then ALL three events can
occur in m  n  p ways.
Steps:
(1) Figure out how many overall decisions you need to make. This is the number of blanks you need
to write down. Label each blank with the kind of decision it is. (who presents 1st, pizza crust, etc)
(2) In each blank, write the number of choices you have within that category. This number will never
be zero.
(3) When all the blanks are filled, multiply the numbers together to get your answer
2
Key Info: _____ Digits
______ Letters in the alphabet
Examples:
1.) Elaine needs to do laundry! She has only 2 clean shirts and 3 clean pants left in her closet. How
many shirt-pant combinations are available to her to wear today?
2.) There are 3 trails on the north face of Mount Ezra and 2 trails on the south face of Mount Ezra.
How many routes are there going up the north face and down the south face?
3.) Kelly must buy rolls for her dinner party. She can buy them in one of 4 nearby supermarkets or
one of 3 local bakeries. In how many ways can Kelly run her errand?
4.) The new iPad is available in Wi-Fi or Wi-Fi 3G, and with 16GB, 32GB, or 64GB. How many
different iPads are possible?
5.) A yogurt shop offers 3 different sizes, 10 flavors, and 8 toppings. How many different orders are
possible if a person orders one size, one flavor, and one topping?
6.) A student council has 5 seniors, 4 juniors, 2 sophomores, and 2 freshmen as members. In how
many ways can a 4-member council committee be formed that includes one member of each class?
7.) In how many different ways can an 8-question true/false test be answered? (Assume that each
question is answered)
8.) In how many ways can an 8-question multiple choice test be answered if each question has five
possible answer choices (a – e)? (Assume that each question is answered)
9.) How many passwords of 3 symbols can be made if each password must start with 2 digits and end
with a letter if (a) all letters and digits can be repeated (b) neither the letters nor the digits can be
repeated?
3
10.) An ice cream shop offers a choice of two types of cones and 15 flavors of ice cream.
a. How many different 2-scoop ice cream cones can a customer order if the same flavor may
not be repeated?
b. How many different 2-scoop ice cream cones can a customer order if the same flavor may
be repeated?
11.) For a college application, Susan must select one of five topics on which to write a short essay. She
must also select a different topic from the same list for a longer essay. How many ways can she choose
the topics for the two essays?
12.) Carla wants to take 6 different classes next year. Assuming that each class is offered each period
and there are 12 classes to choose from, how many different schedules could she have?
13.) In 2011, 572 seniors will graduate from CCA. In how many different orders can these seniors
walk at graduation?
Factorial Definitions:
n !  n(n  1)(n  2)  3  2 1
0!  1
PERMUTATIONS
When a group of objects are arranged in a certain order, the arrangement is called a permutation.
Think of permutations as arrangements of objects in which a change in order would result in a new
arrangement or permuation.
Examples of permutations:
Examples of groupings that are NOT permutations:
If a set has n objects, then the number of ways of ordering r objects from the set is denoted by n Pr
n!
n Pr 
(n  r )!
Examples: Find the number of permutations.
14.) 7 P5
15.) 14 P12
4
16.) 6 P2
17.) 4 P4
Use permutations and the fundamental counting principle to solve:
18.) The Canyon Crest Academy basketball team is in a tournament with 5 other teams. In how many
different ways can the 5 teams finish the tournament?
19.) A baseball team has nine players. How many different batting orders are possible assuming that
every player will be allowed to bat?
20.) Eight people enter a cookie contest at the county fair. How many ways can the 1st, 2nd, and 3rd
place ribbons be awarded?
21.) A club has nine members. In how many ways can a president, vice president, treasurer, and
secretary be chosen from the members of this club?
22.) In how many ways can 6 different books be arranged on a shelf?
23.) From a set of 9 different books, 4 are to be selected and arranged on a shelf. How many different
arrangements are possible?
24.) Listing them all out, how many different arrangements are there of the letters E, A, and T?
25.) Listing them all out, how many different arrangements are there of the letters M, O, and M?
Both #24 and #25 asked you to rearrange 3 letters? Why are the answers different?
5
If objects in the set, being arranged in a specific order, are considered the same/identical,
then the problem is considered to be a permutation with repetition.
Permutations with Repetitions:
The number of distinguishable permutations of n objects of which p are identical and q are identical,
n!
and b are alike, is
p !q !b !
Example: Find the number of distinguishable permutations of the letters in each word.
26.) GEOMETRY
27.) MISSISSIPPI
28.) MATHEMATICS
Examples:
29.) Find the number of different ways of placing 15 balls in a row given that 4 are red, 3 are yellow,
6 are black, & 2 are blue.
30.) Fourteen construction workers are to be assigned to three different tasks. Seven workers are
needed for mixing cement, five for laying bricks, & two for carrying bricks. In how many
different ways can the workers be assigned to these tasks?
31.) A store clerk is setting up a window display. In how many ways can he arrange 5 shirts, 3 purses,
& 7 pairs of shoes if each type of item is seen as being identical?
32.) In how many different ways can 3 identical emeralds, 2 identical diamonds, and 2 different opas
be arranged in a row in a display case?
HW #26—pg. 686 #10-28even, 30-39x3, 44, 46, 50 and Ch. 10 Worksheet #1
6
Notes #27— Using Combinations & the Binomial Theorem (Section 10.2)
COMBINATIONS:
An arrangement or selection of objects in which order is not important is called a combination.
Think of a combination as a grouping.
Examples of combinations (groups in which arrangement order is not important:
The number of combinations of n distinct objects taken r at a time is given by
n!
n Cr 
(n  r )!r !
Examples: Find the number of combinations.
1.) 10 C6
2.) 8 C3
Examples:
3.) A group of seven students working on a project needs to choose two from their group to present
the group’s report to the class. How many ways can they choose the two students?
4.) You can order a hamburger with cheese, onion, pickle, relish, mustard, lettuce, tomato, or
mayonnaise. How many different hamburgers could you build if you can only choose 4 toppings?
5.) In how many ways can a committee of 6 be chosen from 5 teachers and 4 students if:
a) all are equally eligible?
b) the committee must include 3 teachers and 3
students?
6.) Your Algebra 2 class has 4 seniors, 10 juniors, 22 sophomores, and 2 freshmen. You need to form
a class committee of 1 senior, 2 junior, 3 sophomores, and 1 freshman. How many such committees
are possible? (Start with the blanks, then fill them in)
7
Key Info: Deck of Cards
Total
4 Suits
13 cards in each suit
Diamonds
52 cards
3 face cards: King, Queen, Jack
Hearts
Spades
10 non-face cards: Ace – Ten
Clubs
- How many diamonds are in a deck? _______
- How many red cards are in a deck? ______
- How many black face cards are in a deck? ______ - How many sevens are in a deck? _______
7.) How many possible 5-card hands consist of all red cards?
8.) How many possible 5-card hands consist of all face cards?
9.) Five cards are drawn from a standard deck of cards. How many possible hands consist of three
clubs and two diamonds?
10.) Six cards are drawn from a standard deck of cards. How many possible hands consist of two hearts
and four spades?
11.) 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. How many sets of exactly 3 tragedies and 2 histories
can you read?
8
Examples: Determine whether each situation involves a permutation or combination. Then solve!
12.) Choosing 2 different pizza toppings from a list of 6.
13.) Arranging seven shoppers in line at a checkout counter.
14.) Creating “words” out of the letters in the word “intercept”.
15.) How many ways can 9 books be arranged on a shelf?
16.) You have enough money to buy 3 books out of the 20 that are on sale. How many different sets
of three books can you buy?
17.) Choosing a hand of seven cards from a standard deck of cards consisting of five cards from one
suit and two card from another suit.
Pascal’s Triangle:
9
Binomial Theorem:
For any positive integer n, the binomial expansion of  a  b  is:
n
 a  b n n C0a nb0  n 1 C1a n 1b1  n  2 C2a n  2b2  ...  n Cn a0bn
The coefficients of Pascal’s triangle will REPLACE the n C r terms in the expansion of a
binomial.
Examples: Use the binomial theorem to write the binomial expansion.
11.)
 x  35
12.)
 a  2b 4
13.)  3 x  2 y 
5
Examples: Find the coefficient indicated.
14.) Find the coefficient of x 4 in the expansion of (2 x  3)7
15.) Find the coefficient of x6 in the expansion of ( x  2 y )8
16.) Find the coefficient of x3 in the expansion of (2 x  5)6
HW #27: pg. 687: 7-17odd ; pg. 694: 2-10even, 13-18all, 25-31odd, 32-34all; Wkst #2 (hints )
10
Notes #28 — Define & Use Probability (Section 10.3)
An event is…
Definition of probability:
number of outcomes in the event
Probability 
total number of outcomes possible
There are 10 marbles in a bag: 4 are red, 3 are blue, 2 are green, and 1 is yellow.
1.) If you choose one marble at random, what is
2.) If you choose one marble at random, what is
the probability that it is red?
the probability that it is red or blue?
3.) If you choose two marbles in sequence
without replacement, what is the probability that
you choose a blue and then a green?
4.) If you choose two marbles in sequence with
replacement, what is the probability that you
choose a yellow and then a red?
5.) If you choose two marbles at the same time,
what is the probability that you choose a red and
yellow?
6.) If you choose two marbles at the same time,
what is the probability that they are both blue?
7.) If you choose three marbles at the same time, 8.) If you choose two marbles at the same time,
what is the probability that you choose 2 red and 1 what is the probability that they are both yellow?
blue?
Consider the word COMPUTE.
9.) If you choose one letter at random, what is the
probability that it is a vowel?
11.) If you choose two letters at the same time,
what is the probability that one is a vowel and one
is a consonant?
10.) If you choose two letters at the same time,
what is the probability that they are both
consonants?
12.) If you choose three letters at the same time,
what is the probability that they are all
consonants?
11
Consider a standard deck of cards.
13.) You choose one card at random, what is the
probability that it is a face card?
14.) You choose one card at random.
Find P(the card is a five or a six)
15.) You choose one card at random.
Find P(the card is a heart or a spade)
16.) You choose two cards at random. Find
P(the cards are both hearts)
17.) A five-card poker hand is drawn. What is
the probability that all five cards are black?
18.) A five-card poker hand is drawn. What is
the probability that three cards are aces and two
cards are kings?
19.) A five-card poker hand is drawn. What is
the probability that you have four queens?
20.) A five-card poker hand is drawn. What is
the probability that you have four of a kind?
21.) A five-card poker hand is drawn. What is the
probability that all five cards are spades?
22.) A five-card poker hand is drawn. What is
the probability that all five cards are from the
same suit?
Emily has 16 bangle bracelets: 7 silver, 4 gold, 3 with crystals, and 2 with charms.
23.) Emily selects one at random to wear on
24.) Emily selects two at random to wear on
Monday. What is the probability that she chooses Tuesday. What is the probability that she chooses
a gold bracelet?
two crystal bracelets?
25.) Emily selects three at random to wear on
Wednesday. What is the probability that she
chooses two gold bracelets and one silver
bracelet?
26.) Emily selects four at random to wear on
Thursday. What is the probability that she
chooses three silver bracelets and one bracelet
with charms?
12
A case of tennis balls contains a dozen cans of 3 tennis balls. 5 tennis balls in the case are
defective and don’t bounce.
27.) If one ball is selected at random, what is the 28.) If one ball is selected at random, what is the
probability that it is defective?
probability that it is not defective?
29.) If two balls are chosen at random, what is the 30.) If two balls are chosen at random, what is the
probability that both are defective?
probability that neither are defective?
31.) If three balls are chosen at random, what is
the probability one is defective and two are not
defective?
32.) If five balls are chosen at random, what is
the probability that two are defective?
Review: For #33-36, identify whether each represents a permutation or combination and then
solve. For #37-38, use the binomial theorem.
33.) The number of arrangements of the letters in 34.) The number of ways that a 3-person
the word “TANGERINE”
committee can be formed from a group of 10
people.
35.) The number of ways one can get 4-of-a-kind
(in a 5-card hand) while playing poker.
36.) The number of ways that 12 runners can
finish 1st, 2nd, and 3rd in a race.
37.) Find the 5th term of  2 x  3 y 
38.) Find the y 6 term of  x  4 y 2 
7
5
HW #28—pg. 701 #3-16all and Ch. 10 Worksheet #3
13
Notes #29—Find Probabilities of Disjoint & Overlapping Events (Section 10.4)
Compound (“OR”) Probability Problems:
If A and B are events, then the probability of A OR B is:
P(A or B) = P(A) + P(B) – P(A and B)
Two events cannot occur at the same time, they are called mutually exclusive or disjoint, therefore
P(A and B) = 0.
Consider a standard deck of 52 cards. One card is drawn at random. Find each probability.
1.) P(ace or 5)
2.) P(spade or 2)
3.) P(heart or face card)
4.) P(heart and face card)
5.) P(7 or a queen)
6.) P(7 and a queen)
A 6-sided die is rolled. Find each probability.
7.) P(even)
8.) P(1 or 6)
9.) P(even or prime)
10.) P(multiple of 2 or multiple of 3)
11.) P(multiple of 2 and multiple of 3)
12.) P(multiple of 7)
Ten slips of paper are placed in a container. Each is labeled with a number from 1 through 10.
One slip of paper is chosen at random. Find each probability of picking each of the following
from the container:
13.) P(1 or 10)
14.) P(3 or odd)
15.) P(factor of 6)
16.) P(factor of 6 and factor of 8)
14
Two marbles are simultaneously drawn at random from a bag containing 3 red, 5 blue, and 6
green marbles. Find each probability.
17.) P(at least one red marble)
18.) P(two marbles of the same color)
19.) P(two marbles of different colors)
20.) P(at least one green marble)
21.) P(exactly one green marble)
22.) P(no green marbles)
Two six-sided die are rolled. Find each probability:
17.) P(the sum is 8)
18.) P(you roll doubles)
19.) P(the sum is not 6)
20.) P(the sum is less than 5 or greater than 10)
HW 29 —pg. 710 #20-25all, 35-38all and Ch. 10 Worksheet #4
15
Notes #30—Find Probabilities of Independent and Dependent Events (Section 10.5)
Two events are independent if the outcome of the first event does not affect the outcome
of the 2nd event.
Two events are dependent if the outcome of the first event does affect the outcome of
the 2nd event.
Independent/Dependent (“AND”) Probability Problems:
If A and B are events, then the probability of A and B occurring is:
P ( A and B)  P( A)  P( B)
Examples: Two cards are drawn from a standard deck of cards one at a time. Find each probability if
no replacement occurs.
1. P(jack, jack)
2. P(heart, club)
3. P(two diamonds)
4. P(2 of hearts, diamond)
5. P(2 red cards)
6. P(2 black aces)
Examples: Find each probability.
7.) Monica has 7 blue pens, 3 black pens, and 2 red pens in her desk. If she selects three pens at
random with no replacement, what is the probability that she will first select a blue pen, then a
black pen, and then another blue pen?
8.) A contestant on a game show reaches into a container without looking and picks two paper bills.
There are 2 $100 bills, 4 $50 bills, 10 $20 bills, and 20 $10 bills. What is the probability that the
contestant draws 2 $10 bills one after the other without replacement?
16
9.) Find the probability of drawing a face card and then an ace from a standard 52 card deck (a) with
replacement and (b) without replacement.
Mixed Review:
Sara has 9 rings to choose from. 3 are silver, 2 are gold, 4 are plastic. If the rings are chosen at
random, find each probability:
11.) Three rings are chosen. Find: P(2 gold, 1
10.) Three rings are chosen. Find: P(all silver)
plastic)
12.) Two rings are chosen. Find: P(at least one
plastic)
13.) Two rings are chosen. Find: P(none are gold)
14.) One ring is chosen. Find: P(silver or gold)
15.) Three rings are chosen. Find: P(one of each
kind)
Consider a standard deck of 52 cards. Cards are randomly chosen. Find each probability.
16.) One card is chosen. Find: P(spade or king)
17.) One card is chosen. Find: P(heart or face
card)
18.) One card is chosen. Find: P(heart and face
card)
19.) Five cards are chosen. Find: P(three jacks)
20.) Five cards are chosen. Find: P(three of a
kind)
21.) Five cards are chosen. Find: P(full house)
(three of a kind and two of a different kind)
17
Solve.
23.) Eight students tried out for the three parts the
22.) When getting dressed today, you had the
choice of wearing six sweaters, five pairs of jeans, school play. In how many different ways can they
be cast?
and 3 pairs of shoes. How many different outfits
were possible?
24.) You have created a 12-song playlist on your
iPod. In how many different ways can you play
these 12 songs on shuffle?
25.) You are giving three of your friends a ride
home from the volleyball game. In how many
different ways can you take them home?
26.) Your mom says that you need to clean your
room, do the dishes, and take out the trash before
you can play your PS3. In how many different
ways can you complete your chores?
27.) You have just won a reward challenge on
Survivor and you can choose 2 of the other
remaining 8 players to come with you on the
helicopter ride. How many different groups could
attend the reward challenge?
28.) You are taking a 4-question, multiple choice,
homework quiz in your Algebra 2 class. Each
question has answer choices A, B, C, D, and E. In
how many different ways can you answer these
questions?
29.) Let’s say you have absolutely NO idea how
to solve any of those four questions on the
homework quiz and you randomly guess on all
four questions. What is the probability that you
get a 100% on your quiz?
HW 30— pg. 721#9-14all, 26-30 even and Ch. 10 Worksheet #5
18