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Objective
Find the number of combinations of
objects
Vocabulary
Combination
An arrangement or listing in which order is not
important
C(a, b) = P(a, b)
b!
Math Symbols
C(a, b)
The number of combinations of a things taken
b at a time
Example 1 Use a Combination
Example 2 Find a Combination Notation
Example 3 Combinations and Permutations
Example 4 Combinations and Permutations
Find
Write the combination statement
A combination is another
modified permutation and order
is not important
Write formula for combination
using information in combination
statement
The numerator is a permutation
of the statement
The denominator is a factorial of
the second number
1/4
P(8, 5) is a permutation that
begins with 8 and multiply 5
descending numbers
Find
Write definition of 5!
C(8, 5) =
87654
54321
C(8, 5) = 6,720
120
5! is a permutation that begins
with 5 and multiply all
descending numbers to 1
Follow Order of Operations
P
E
MD
AS
Multiply in numerator
Multiply in denominator
1/4
Find
C(8, 5) =
87654
54321
Divide numerator by
denominator
C(8, 5) = 6,720
120
Answer:
C(8, 5) = 56
1/4
Find
Answer: C(6, 3) = 20
1/4
TOURNAMENTS Five teams are playing each other in a
tournament. If each team plays every other team once,
how many games are played?
C(a, b) = P(a, b)
b!
Order does not matter so this is
a combination
C(5,
Write the combination statement
Write formula for combination
“a” represents the number of
choices
Replace a with 5
2/4
TOURNAMENTS Five teams are playing each other in a
tournament. If each team plays every other team once,
how many games are played?
C(a, b) = P(a, b)
b!
C(5, 2) =
P(5, 2)
2!
“b” represents the number wants
to choose
“each team plays every other
team” refers to 2 teams playing
each other at a time
Replace b with 2
Complete the equation by
replacing values for a and b
2/4
TOURNAMENTS Five teams are playing each other in a
tournament. If each team plays every other team once,
how many games are played?
C(a, b) = P(a, b)
b!
P(5, 2)
2!
C(5, 2) = 5  4
21
C(5, 2) =
P(5, 2) is a permutation that
begins with 5 and multiply only 2
numbers
Write definition of 2!
2! is a permutation that begins
with 2 and multiply all
descending numbers to 1
2/4
TOURNAMENTS Five teams are playing each other in a
tournament. If each team plays every other team once,
how many games are played?
C(a, b) = P(a, b)
b!
P(5, 2)
2!
C(5, 2) = 5  4
21
C(5, 2) = 20
2
Answer:
C(5, 2) = 10 games
C(5, 2) =
Follow Order of Operations
P
E
MD
AS
Multiply in numerator
Multiply in denominator
Divide numerator by
denominator
Add dimensional analysis
2/4
TOURNAMENTS Six teams are playing each other in a
tournament. If each team plays every other team once,
how many games are played?
Answer: C(6, 2) = 15 games
2/4
SCHOOL An eighth grade teacher needs to select
4 students from a class of 22 to help with sixth grade
orientation. Does this represent a combination or a
permutation? How many possible groups could be
selected to help out the new students?
Order does not matter so this is
P(a, b)
C(a, b) =
a combination
b!
Write the combination statement
C(22,
Write formula for combination
“a” represents the number of
choices
Replace a with 22
3/4
SCHOOL An eighth grade teacher needs to select
4 students from a class of 22 to help with sixth grade
orientation. Does this represent a combination or a
permutation? How many possible groups could be
selected to help out the new students?
C(a, b) =
P(a, b)
b!
C(22, 4) =
P(22, 4)
4!
“b” represents the number wants
to choose
Replace b with 4
Complete the equation by
replacing values for a and b
3/4
C(a, b) = P(a, b)
b!
C(22, 4) =
P(22, 4)
4!
P(22, 4) is a permutation that
begins with 22 and multiply only
4 numbers
Write definition of 4!
Follow Order of Operations
C(22, 4) = 22  21  20  19
4321
C(22, 4) = 175,560
24
P
E
MD
AS
Multiply in numerator
Multiply in denominator
Divide numerator by
denominator
C(22, 4) = 7,315
3/4
C(a, b) = P(a, b)
b!
C(22, 4) =
P(22, 4)
4!
Add dimensional analysis
How many possible groups
could be selected to help out
the new students?
C(22, 4) = 22  21  20  19
4321
C(22, 4) = 175,560
24
Answer:
C(22, 4) = 7,315 groups
3/4
SCHOOL A teacher needs to select 5 students from a
class of 26 to help with parent teacher conferences.
Does this represent a combination or a permutation?
How many possible groups could be selected to help?
Answer: C(26, 5) = 65,780 groups
3/4
SCHOOL An eighth grade teacher needs to select
4 students from a class of 22 to help with sixth grade
orientation. One eighth grade student will be assigned to
sixth grade classes on the first floor, another student will
be assigned to classes on the second floor, another
student will be assigned to classes on the third floor, and
still another student will be assigned to classes on the
fourth floor. Does this represent a combination or a
permutation? In how many possible ways can the eighth
graders be assigned to help with the sixth grade
orientation?
Shows order is important
P(a, b) =
Write permutation formula
4/4
SCHOOL An eighth grade teacher needs to select
4 students from a class of 22 to help with sixth grade
orientation. One eighth grade student will be assigned to
sixth grade classes on the first floor, another student will
be assigned to classes on the second floor, another
student will be assigned to classes on the third floor, and
still another student will be assigned to classes on the
fourth floor. Does this represent a combination or a
permutation? In how many possible ways can the eighth
graders be assigned to help with the sixth grade
orientation?
“a” represents the number of
P(a, b) =
P(22,
choices
Replace a with 22
4/4
SCHOOL An eighth grade teacher needs to select
4 students from a class of 22 to help with sixth grade
orientation. One eighth grade student will be assigned to
sixth grade classes on the first floor, another student will
be assigned to classes on the second floor, another
student will be assigned to classes on the third floor, and
still another student will be assigned to classes on the
fourth floor. Does this represent a combination or a
permutation? In how many possible ways can the eighth
graders be assigned to help with the sixth grade
orientation?
“b” represents the number wants
to choose
P(a, b) =
P(22, 4) =
Replace b with 4
4/4
P(a, b) =
P(22, 4) is a permutation that
begins with 22 and multiply 4
numbers
P(22, 4) = 22  21  20  19
Multiply
Answer:
P (22, 4) = 175,560 ways
Add dimensional analysis
In how many possible ways
can the eighth graders be
assigned to help with the
sixth grade orientation?
4/4
*
SCHOOL A teacher needs to select 5 students from a
class of 26 to help with parent teacher conferences.
One student will be assigned to fifth grade parents,
another student will be assigned to sixth grade parents,
another student will be assigned to seventh grade
parents, another student will be assigned to eighth
grade parents, and still another student will be
assigned to ninth grade parents. Does this represent a
combination or a permutation? In how many possible
ways can the students be assigned to help with the
parent teacher conferences?
Answer: P(26, 5) = 7,893,600 ways
4/4
Assignment
Lesson 8:4
Combinations
3 - 26 All