unit 9 mid checkpoint apply, algebra homework help

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Cassidy Williams
Assessments
Unit 9: Applications of Probability Mid Checkpoint Apply
Unit 9: Mid Checkpoint Apply
Interactive Tools
Unit 9: Applications of Probability Mid Checkpoint Apply
Lesson 1 – 3
Please complete the following questions. It is important that you show all work you
did to solve the problems when you submit your work. This includes any
calculations, diagrams, or graphs that helped you solve it.
Algebra 2
1. OPEN ENDED Describe a situation in which the number of outcomes is given by
26,3).
2. OPEN ENDED Describe an event that has a probability of O and an event that
has a probability of 1.
3. Writing in Math Use the information on page 684 to explain how you can count
the maximum number of license plates a state can issue. Explain how to use the Fundamental Counting Principle to
find the number of different license plates in a state such as Oklahoma, which has 3 letters followed by 3 numbers.
Also explain how a state can increase the number of possible plates Without increasing the length of the plate
number.
4. Writing in Math Use the information on page 710 to explain how permutations and combinations apply to softball.
Explain how to find the number of 9-person lineups that are possible and how many ways there are to choose 9
players if 16 players show up for a game.
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Write two more rows of Pascal’s triangle. Then use the patters of
the cients Write
American
High School
LA
Main Ideas
. Uwcastlange
the and post
binomas
. Uw the inmal
Theorm to espand
powers of biomis
New Vocabulary
Paulstring
Binomial Theorem
factorial
684 – 685 / 1100
GET READY for the Lesson
According to the US Census Bureau, ten percent of families have
three or more children. Il a family has four children, there are six
sequences of births of boys and girls that result in two boys and
two girls. These sequences are listed below
DIGG
TXGB GANG ССП GGDE
(* + y = 1x?y + 7x*y+ 21×2 + xy 100%
= x? +7xy + 2x +354
CHECK Your Progress
1. Expand ()
TCRC .
The Binomial Theorem Another way slow
cxpansion is to write them in terms of the previa
(9+)
-Pascal’s Triangle You can use the coefficients in powers of binomials
to count the number of possible sequences in situations such as the one
above. Expand a few powers of the binomials +
() + 8)0 –
147
15+ x) =
1+14
(1 + x)2 =
153 +211 + 1/2
1 +81′
1578″ + 3871 +3818+10
() + 19 –
1543″ this’ ++48 g + 1
1
2:1
(a+b
1
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2
1:3
3.2.1
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1
4
4.3.2
Study Tip
This pattern
summarized in the Binomial Theorem
Terms
The open of :
Drama teha
po 2-1
Sorms. For enke
10 by has 7 term.
KEY CONCEPT
Binomial Theorem
in is a nonnegative integer, then (a + b)” – 10″-16
2 g 32+ ” 207 393 + … + 10%
The ethicient 4 of the big term in the expansion of (b + 1 gives the
number of sequences of births that result in one boy and three girls
Here are some patterns in any binomial expansion of the form (a + b)”.
1. There are n + 1 terms.
2. The expectent of a + ” is the exponent of a in the first term and
the exponent of it in the last team
3. In successive terms, the exponent of a decreases by one, and the
exponent of increases by one.
4. The sum of the exponents in each terris
5. The coefficients are symmetric. They increase at the beginning of the
expansion and decrease at the end
The coefficients form a pattern that is often displayed in a triangular
tormation. This is known as Pascal’s triangle. Notice that each row
begins and ends with 1. Each coefficient is the sum of the two coefficients
above it in the previous row.
a + b)
1 + b)
1a + b)2
Real World Unh
Nthough he ad not
GREAT I, Pascats
trangle is named
for the Free
mathematician Blaise
Pascal (1623-15621.
Study Tip
Coefficients
Matat ns
haaing the same
esponse
reverse, as in 15
and 1500
EXAMPLE Use the Binomial Theorem
Expand la-6)
Use the sequence 1, 11:9.5 to find the coefficients for the first four
terms. Then use symmetry to find the remaining cetticients.
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CHECK Your Progress
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High School
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Main Ideas
. Suke problems
Prmindful
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cominans
(ID – 3
–
GET READY for the Lesson
When the manager of a softball team ills
out her team’s lineup card before the game,
the order in which she fills in the names is
important because it determines the order
in which the players will bat.
Suppone she has 7 posible players in mind
for the top 4 spots in the lineup. You know
from the fundamental Counting Principle
that there are 7.6.5.4 or 540 ways that
she could assign players to the top 4 spots
The gold, silver, and bronze medas can b4100%nded in 20 wa
CHECK Your Progress
1. A newspaper has nine reporters available
How many ways can the texts be assig
-9.9.20
New Vocabulary
permutation
Suppose you want to rearrange the tters of the
can make a different word. If the two es were
the word could le arranged in P 8.8 ways.
The
divide 48,8) by the number of arrangements
in P(2.2) ways
combination
18,81
S!
M12, 273
8.7.6-5.-1.3.2!
or 20,160 Simplify
Thus, there are 27,160 ways to arrange the Ictters in gemetry.
Permutations When a group of objects or people are arranged in a
certain order the arrangement is called a permutation. In a permutation,
the order of the objects is very important. The arrangement of objects or
prople in a line is called a lincar permutation.
Notice that 7.6.5.4 is the product of the first 4 factors of 71. You can
rewrite this product in terms of 1
7.6.5.4 = 7.6.3.4.3.2.1
3.2.1
When some letters or objects are alike, we the rule below to find the num
of permutations
KEY CONCEPT
Permutations with Repetiti
The number of permutations of n objects of which p are alike and gare alike is
= 7.6-5.4:3.2.1 or 21-7-6-5-4-3-2-1 and 31 -3.2.1
pligt
This rule can be extended to any number of objeds that are repeated
Notice that is the same 17-4)
The number of ways to arrange 7 people or objects taken 4 at a time is
written “[7, 4). The expression for the sotthall lineup above is a case of
the following formula
KEY CONCEPT
Permutations
The number of pemutations of a distinct objects taken at a time is given by
Pinn) –
ni
(n-1)
Reading Math
Permutations the
spression in reads
the number of
PAVOL
culars tolawa
be is sometimes
EXAMPLE Permutation with Repetition
How many different ways can the letters of the word MISSISSIPPI be
arranged?
The letter occurs 4 times, Soccurs 4 times, and P occurs wie
You need to find the number of permutations of 11 letters of which 4 of one
letter, 4 of another latter, and 2 of another Ictter and the same.
11.10.9.4.2.6.5.40 34,650
1121
There are 34,650 ways arrange the letters.
CHECK Your Progress
2. How many different ways can the letters of the word DECIDED be
weena
EXAMPLE Permutation
FIGURE SKATING There are 10 finalists in a figure skating competition.
How many ways can gold, silver and bronze medals be awarded?
Since each winner will recive a diferent medal ander is important. You
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