Transcript Combinations - Skyline R2 School

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Objective
Find the number of combinations of a set of
objects
Vocabulary
Combination
An arrangement, or listing, of objects in which
order is not important
C(a, b) = P(a, b)
b!
Vocabulary
Permutation
An arrangement, or listing, of objects in which
order is important
P(a, b) = P(a, b)
Example 1 Find the Number of Combinations
Example 2 Use a Combination to Solve a Problem
Example 3 Identify Permutations and Combinations
Example 4 Identify Permutations and Combinations
DECORATING Ada can select from seven paint colors
for her room. She wants to choose two colors. How
many different pairs of colors can she choose?
C(a, b) =
She wants to choose 2 colors
which makes this a combination
Write the combination statement
A combination is another
modified permutation and order
is not important
1/4
DECORATING Ada can select from seven paint colors
for her room. She wants to choose two colors. How
many different pairs of colors can she choose?
C(a, b) = P(a, b)
b!
C(7, 2) =
Write formula for combination
“a” represents the number of
choices
Replace a with 7
“b” represents the number wants
to choose
Replace b with 2
1/4
DECORATING Ada can select from seven paint colors
for her room. She wants to choose two colors. How
many different pairs of colors can she choose?
C(a, b) =
P(a, b)
b!
C(7, 2) =
P(7, 2)
2!
C(7, 2) = 7  6
21
Complete the equation by
replacing values for a and b
P(7, 2) is a permutation that
begins with 7 and multiply only 2
numbers
Write definition of 2!
1/4
DECORATING Ada can select from seven paint colors
for her room. She wants to choose two colors. How
many different pairs of colors can she choose?
C(a, b) =
C(7, 2) =
P(a, b)
b!
P(7, 2)
2!
C(7, 2) = 7  6
21
C(7, 2) = 42
2
Answer:
C(7, 2) = 21 pairs of colors
Follow Order of Operations
P
E
MD
AS
Multiply in numerator
Multiply in denominator
Divide numerator by
denominator
Add dimensional analysis
1/4
HOCKEY The Brownsville Badgers hockey team has
14 members. Two members of the team are to be
selected to be the team’s co-captains. How many
different pairs of players can be selected to be the
co-captains?
Answer: C(14, 2) = 91 pairs of players
1/4
INTRODUCTIONS Ten managers attend a business
meeting. Each person exchanges names with each other
person once. How many introductions will there be?
P(a, b)
C(a, b) =
b!
C(10, 2) =
Write the combination formula
“a” represents the number of
choices
Replace a with 10
“b” represents the number wants
to choose
This represents 2 people
Replace b with 2
2/4
INTRODUCTIONS Ten managers attend a business
meeting. Each person exchanges names with each other
person once. How many introductions will there be?
P(a, b)
C(a, b) =
b!
Complete the equation by
replacing values for a and b
P(10, 2)
C(10, 2) =
2!
C(10, 2) = 10  9
21
P(10, 2) is a permutation that
begins with 10 and multiply only
2 numbers
Write definition of 2!
2/4
INTRODUCTIONS Ten managers attend a business
meeting. Each person exchanges names with each other
person once. How many introductions will there be?
P(a, b)
Follow Order of Operations
C(a, b) =
b!
P
E
MD
AS
P(10, 2)
C(10, 2) =
2!
Multiply in numerator
10  9
C(10, 2) =
Multiply in denominator
21
Divide numerator by
C(10, 2) = 90
2
denominator
Add dimensional analysis
Answer:
C(10, 2) = 45 introductions
2/4
PHYSICAL EDUCATION Twelve students in a
physical education class must pair off for a particular
exercise. How many different pairs are possible?
Answer: C(12, 2) = 66 pairs
2/4
TRACK From an eight-member track team, three
members will be selected to represent the team at the
state meet. How many ways can these three members
be selected? Does the situation represent a
permutation or a combination?
Combination
P(a, b)
C(a, b) =
b!
Combination = order not important
Permutation = order important
Determine if order is important
Order is not important because a
member can be in any selection
Write combination formula
3/4
TRACK From an eight-member track team, three
members will be selected to represent the team at the
state meet. How many ways can these three members
be selected? Does the situation represent a
permutation or a combination?
“a” represents the number of
Combination
choices
P(a, b)
C(a, b) =
Replace a with 8
b!
C(8, 3) =
“b” represents the number wants
to choose
Replace b with 3
3/4
TRACK From an eight-member track team, three
members will be selected to represent the team at the
state meet. How many ways can these three members
be selected? Does the situation represent a
permutation or a combination?
Complete the equation by
Combination
replacing values for a and b
P(a, b)
C(a, b) =
b!
P(8, 3) is a permutation that
P(8, 3)
C(8, 3) =
begins with 8 and multiply 3
3!
numbers
C(8, 3) = 8  7  6
321
Write definition of 3!
3/4
Follow Order of Operations
Combination
P
E
MD
AS
C(a, b) = P(a, b)
b!
Multiply in numerator
C(8, 3) = P(8, 3)
3!
Multiply in denominator
C(8, 3) = 8  7  6
321
C(8, 3) = 336
6
Answer:
C(8, 3) = 56 ways
Divide numerator by
denominator
Add dimensional analysis
How many ways can these
three members be selected?
3/4
*
COMMITTEES In how many ways can you choose a
committee of four people from a staff of ten? Does
the situation represent a permutation or a
combination? Solve the problem.
Answer: Combination
C(10, 4) = 210 ways
3/4
TRACK In how many ways can you choose the first,
second, and third runners in a relay race from eight
members of a track team? Does the situation represent a
permutation or a combination?
P(a, b) = P(a, b)
Combination = order not important
Permutation = order important
Determine if order is important
Order is important because a
runner in first cannot run in
second
Write permutation formula
4/4
TRACK In how many ways can you choose the first,
second, and third runners in a relay race from eight
members of a track team? Does the situation represent a
permutation or a combination?
P(a, b) = P(a, b)
P(8, 3) = P(8, 3)
“a” represents the number of
choices
Replace a with 8
“b” represents the number wants
to choose
Replace b with 3
Complete the equation by
replacing values for a and b
4/4
TRACK In how many ways can you choose the first,
second, and third runners in a relay race from eight
members of a track team? Does the situation represent a
permutation or a combination?
P(8, 3) = P(8, 3)
P(8, 3) is a permutation that
begins with 8 and multiply 3
numbers
P(8, 3) = 8  7  6
Multiply
P(a, b) = P(a, b)
Answer:
P(8, 3) = 336 ways
Add dimensional analysis
4/4
PTA There are fifteen members on the PTA for a local
middle school. Three of those fifteen will be elected
for the offices of president, secretary, and treasurer
of the PTA. How many ways can these three positions
be filled? Does the situation represent a permutation
or a combination? Solve the problem.
Answer: Permutation
P(15, 3) = 2,730 ways
4/4
Assignment
Lesson 9:5
Combinations
4 - 16 All