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Section
.6 Permutations and Combinations 10.6 Permutations and Combinations
In this section, we will learn to
1.Use the Multiplication Principle for Events.
2.Solve permutation problems.
3.Solve combination problems.
4.Use combination notation to write the Binomial Theorem.
5.Find the number of permutations of like things.
1. ​​Use the Multiplication Principle for Events
Lydia plans to go to dinner and attend a movie. If she has a choice of four restaurants and three movies, in how many ways can she spend her evening? There are
four choices of restaurants and, for any one of these choices, there are three choices
of movies, as shown in the tree diagram in Figure 10-3.
Cheesecake Factory
Lydia’s
choices
Outback Steakhouse
P.F. Chang’s
True Grit
Chronicles of Narnia: The Voyage of the Dawn Treader
Black Swan
True Grit
Chronicles of Narnia: The Voyage of the Dawn Treader
Black Swan
True Grit
Chronicles of Narnia: The Voyage of the Dawn Treader
Black Swan
True Grit
Chronicles of Narnia: The Voyage of the Dawn Treader
Black Swan
Not For Sale
Olive Garden
FIGURE 10-3
10-2
Chapter
Not For Sale
Sequences,Series,and Probability
Any situation that has several outcomes is called an event. Lydia’s first event
(choosing a restaurant) can occur in 4 ways. Her second event (choosing a movie)
can occur in 3 ways. Thus, she has 4 ? 3, or 12, ways to spend her evening. This
example illustrates the Multiplication Principle for Events.
Multiplication Principle
for Events
Let E1 and E2 be two events. If E1 can be done in a1 ways, and if—after E1 has
occurred—E2 can be done in a2 ways, then the event “E1 followed by E2 ” can
be done in a1 ? a2 ways.
The Multiplication Principle can be extended to n events.
EXAMPLE 1 Using the Multiplication Principle for Events
If a traveler has 4 ways to go from New York to Chicago, 3 ways to go from Chicago
to Denver, and 6 ways to go from Denver to San Francisco, in how many ways can
she go from New York to San Francisco?
SOLUTION We can let E1 be the event “going from New York to Chicago,” E2 the event “going
from Chicago to Denver,” and E3 the event “going from Denver to San Francisco.”
Since there are 4 ways to accomplish E1 , 3 ways to accomplish E2 , and 6 ways to
accomplish E3 , the number of routes available is
4 ? 3 ? 6 5 72
Self Check 1 If a man has 4 sweaters and 5 pairs of slacks, how many different outfits can he
wear? Now Try Exercise 29.
2. ​​Solve Permutation Problems
Suppose we want to arrange 7 books in order on a shelf. We can fill the first space
with any one of the 7 books, the second space with any of the remaining 6 books,
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© Photos 12/Alamy
The diagram shows that Lydia has 12 ways to spend her evening. One possibility is to eat at the Cheesecake Factory and watch True Grit. Another is to eat at the
Olive Garden and watch Black Swan.
Section
.6 Permutations and Combinations 10-3
the third space with any of the remaining 5 books, and so on, until there is only one
space left to fill with the last book. According to the Multiplication Principle, the
number of ordered arrangements of the books is
7 ? 6 ? 5 ? 4 ? 3 ? 2 ? 1 5 5,040
When finding the number of possible ordered arrangements of books on a shelf, we
are finding the number of permutations. The number of permutations of 7 books,
using all the books, is 5,040. The symbol P 1n, r2 is read as “the number of permutations of n things r at a time.” Thus, P 17, 72 5 5,040.
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EXAMPLE 2 Finding the Number of Permutations
Assume that there are 7 signal flags of 7 different colors to hang on a mast. How
many different signals can be sent when 3 flags are used?
SOLUTION We are asked to find P 17, 32 , the number of permutations (ordered arrangements)
of 7 things using 3 of them. Any one of the 7 flags can hang in the top position on
the mast. Any one of the 6 remaining flags can hang in the middle position, and any
one of the remaining 5 flags can hang in the bottom position. By the Multiplication
Principle, we have
P 17, 32 5 7 ? 6 ? 5 5 210
It is possible to send 210 different signals.
Self Check 2 How many signals can be sent if two of the 7 flags are missing? Now Try Exercise 37.
Although it is correct to write P 17, 32 5 7 ? 6 ? 5, we will change the form of
the answer to obtain a convenient formula. To derive this formula, we proceed as
follows:
P 17, 32 5 7 ? 6 ? 5
5
7?6?5?4?3?2?1
4?3?2?1
5
7!
4!
5
Multiply numerator and denominator by 4 ? 3 ? 2 ? 1.
7!
17 2 32 !
The generalization of this idea gives the following formula.
Formula for P 1 n, r 2
The number of permutations of n things r at a time is given by
P 1n, r2 5
n!
1n 2 r2 !
EXAMPLE 3 Using the Permutation Formula
Find: ​a. ​P 18, 42 b. ​P 1n, n2 c. ​P 1n, 02
SOLUTION We will substitute into the formula for P 1n, r2 and simplify.
Not For Sale
a. P 18, 42 5
8!
8 ? 7 ? 6 ? 5 ? 4!
5
5 1,680
18 2 42 !
4!
10-4
Chapter
Not For Sale
Sequences,Series,and Probability
b. P 1n, n2 5
c. P 1n, 02 5
n!
n!
n!
5
5 n!
5
1n 2 n2 !
0!
1
n!
n!
51
5
1n 2 02 !
n!
Self Check 3 Find: ​a. ​P 17, 52 Now Try Exercise 13.
b. ​P 16, 02 Formulas for P 1 n, n 2 and P 1 n, 0 2
The number of permutations of n things n at a time and n things 0 at a time
are given by the formulas
P 1n, n2 5 n! and P 1n, 02 5 1
EXAMPLE 4Solving a Permutation Problem
Richard Paul Kane/Shutterstock.com
In how many ways can a baseball manager arrange a batting order of 9 players if
there are 25 players on the team?
SOLUTION To find the number of permutations of 25 things 9 at a time, we substitute 25 for n
and 9 for r in the formula for finding P 1n, r2 .
P 1n, r2 5
P 125, 92 5
n!
1n 2 r2 !
25!
125 2 92 !
5
25!
16!
5
25 ? 24 ? 23 ? 22 ? 21 ? 20 ? 19 ? 18 ? 17 ? 16!
16!
5 741,354,768,000
The number of permutations is 741,354,768,000.
Self Check 4 In how many ways can the manager arrange a batting order if 2 players can’t
play? Now Try Exercise 41.
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Parts (b) and (c) of Example 3 establish the following formulas.
Section
Accent on Technology
.6 Permutations and Combinations 10-5
Permutations
A graphing calculator can be used to find P 1n, r2 . For example, let’s consider the
permutation in Example 4 and determine P 125, 92 . The calculator operation that
will be used is nPr.
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To find P 1 25, 9 2 , first enter
25 in your calculator.
Press MATH and move
the cursor to PRB.
Scroll down to 2:nPr and
press ENTER .
Enter 9 on your calculator
and press ENTER .
We see that P 1 25, 9 2 is
equal to 741,354,768,000.
FIGURE 10-4
EXAMPLE 5Solving a Permutation Problem
In how many ways can 5 people stand in a line if 2 people refuse to stand next to
each other?
SOLUTION The total number of ways that 5 people can stand in line is
P 15, 52 5 5! 5 5 ? 4 ? 3 ? 2 ? 1 5 120
To find the number of ways that 5 people can stand in line if 2 people insist
on standing together, we consider the two people as one person. Then there are 4
people to stand in line, and this can be done in P 14, 42 5 4! 5 24 ways. However,
because either of the two who are paired could be first, there are two arrangements
for the pair who insist on standing together. Thus, there are 2 ? 4!, or 48 ways that
5 people can stand in line if 2 people insist on standing together.
The number of ways that 5 people can stand in line if 2 people refuse to
stand together is 5! 5 120 (the total number of ways to line up 5 people) minus
2 ? 4! 5 48 (the number of ways to line up the 5 people if 2 do stand together):
120 2 48 5 72
There are 72 ways to line up the people.
Self Check 5 In how many ways can 5 people stand in a line if one person demands to be first? Now Try Exercise 39.
EXAMPLE 6Solving a Permutation Problem
In how many ways can 5 people be seated at a round table?
SOLUTION If we were to seat 5 people in a row, there would be 5! possible arrangements. However, at a round table, each person has a neighbor to the left and to the right. If each
person moves one, two, three, four, or five places to the left, everyone has the same
neighbors and the arrangement has not changed. Thus, we must divide 5! by 5 to
get rid of these duplications. The number of ways that 5 people can be seated at a
round table is
Not For Sale
5!
5 ? 4!
5
5 4! 5 4 ? 3 ? 2 ? 1 5 24
5
5
10-6
Chapter
Not For Sale
Sequences,Series,and Probability
Self Check 6 In how many ways can 6 people be seated at a round table? Now Try Exercise 43.
The results of Example 6 suggest the following fact.
Circular Arrangements
There are 1n 2 12 ! ways to arrange n things in a circle.
Suppose that a class of 12 students selects a committee of 3 persons to plan a party.
With committees, order is not important. A committee of John, Maria, and Raul
is the same as a committee of Maria, Raul, and John. However, if we assume for
the moment that order is important, we can find the number of permutations of
12 things 3 at a time.
P 112, 32 5
12!
12 ? 11 ? 10 ? 9!
5 1,320
5
112 2 32 !
9!
However, since we do not care about order, this result of 1,320 ways is too large.
Because there are 6 ways 13! 5 62 of ordering every committee of 3 students, the
result of P 112, 32 5 1,320 is exactly 6 times too big. To get the correct number of
committees, we must divide P 112, 32 by 6:
1,320
P 112, 32
5
5 220
6
6
In cases of selection where order is not important, we are interested in combinations, not permutations. The symbols C 1n, r2 and anr b both mean the number of
combinations of n things r at a time.
If a committee of r people is chosen from a total of n people, the number of
possible committees is C 1n, r2 , and there will be r! arrangements of each committee.
If we consider the committee as an ordered grouping, the number of orderings is
P 1n, r2 . Thus, we have
(1) r!C 1n, r2 5 P 1n, r2
Comment
When discussing permutations, order
counts. When discussing combinations,
order doesn’t count.
Formula for C 1 n, r 2
We can divide both sides of Equation 1 by r! to obtain the formula for finding
C 1n, r2 .
n!
n
P 1n, r2
5
C 1n, r2 5 a b 5
r!
r! 1n 2 r2 !
r
The number of combinations of n things r at a time is given by
n!
n
C 1n, r2 5 a b 5
r! 1n 2 r2 !
r
In Exercises 77 and 78, you will be asked to prove the following formulas.
Formulas for C 1 n, n 2 and C 1 n, 0 2
If n is a whole number, then
C 1n, n2 5 1 and C 1n, 02 5 1
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3. ​​Solve Combination Problems
Section
.6 Permutations and Combinations 10-7
EXAMPLE 7Solving a Combination Problem
If Carla must read 4 books from a reading list of 10 books, how many choices does
she have?
SOLUTION Because the order in which the books are read is not important, we find the number
of combinations of 10 things 4 at a time:
C 110, 42 5
10!
10 ? 9 ? 8 ? 7 ? 6!
5
4! 110 2 42 !
4 ? 3 ? 2 ? 1 ? 6!
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5
10 ? 9 ? 8 ? 7
4?3?2
5 210
Carla has 210 choices.
Self Check 7 How many choices would Carla have if she had to read 5 books? Now Try Exercise 49.
Accent on Technology
Combinations
A graphing calculator can be used to find C 1n, r2 . For example, let’s consider the
permutation in Example 7 and determine C 110, 42 . The calculator operation that
will be used is nCr.
To find C 1 10, 4 2 , first enter
10 in your calculator.
Press MATH and move
the cursor to PRB.
Scroll down to 3:nCr and
press ENTER .
Enter 4 on your calculator
and press ENTER .
We see that C 1 10, 4 2 is
equal to 210.
FIGURE 10-5
EXAMPLE 8Solving a Combination Problem
A class consists of 15 men and 8 women. In how many ways can a debate team be
chosen with 3 men and 3 women?
SOLUTION There are C 115, 32 ways of choosing 3 men and C 18, 32 ways of choosing 3 women.
By the Multiplication Principle, there are C 115, 32 ? C 18, 32 ways of choosing members of the debate team:
C 115, 32 ? C 18, 32 5
5
15!
8!
?
3! 115 2 32 ! 3! 18 2 32 !
15 ? 14 ? 13 8 ? 7 ? 6
?
6
6
5 25,480
Not For Sale
There are 25,480 ways to choose the debate team.
10-8
Chapter
Not For Sale
Sequences,Series,and Probability
Self Check 8 In how many ways can the debate team be chosen if it is to have 4 men and 2
women? Now Try Exercise 59.
4. ​​Use Combination Notation to Write the Binomial Theorem
C 1n, r2 5
n!
r! 1n 2 r2 !
gives the coefficient of the 1r 1 12 th term of the binomial expansion of 1a 1 b2 n.
This implies that the coefficients of a binomial expansion can be used to solve problems involving combinations. The Binomial Theorem is restated below—this time
listing the 1r 1 12 th term and using combination notation.
Binomial Theorem
If n is any positive integer, then
n
n
n
n
n
1a 1 b2 n 5 a ban 1 a ban21b 1 a ban22b2 1 c1 a ban2rbr 1 c1 a bbn
0
1
2
r
n
Comment
In the expansion of 1a 1 b2 n, the term containing br is given by
n
a ban2rbr
r
EXAMPLE 9 Using Pascal’s Triangle to Compute a Combination
Use Pascal’s Triangle to compute C 17, 52 .
SOLUTION Consider the eighth row of Pascal’s Triangle and the corresponding combinations:
1
7
21
35
35
21
7
1
7
7
7
7
7
7
7
7
a b a b a b a b a b a b a b a b
0
1
2
3
4
5
6
7
7
C 17, 52 5 a b 5 21.
5
Self Check 9 Use Pascal’s Triangle to compute C 16, 52 . Now Try Exercise 75.
5. ​​Find the Number of Permutations of Like Things
A word is a distinguishable arrangement of letters. For example, six words can be
formed with the letters a, b, and c if each letter is used exactly once. The six words are
abc, acb, bac, bca, cab, and cba
If there are n distinct letters and each letter is used once, the number of distinct
words that can be formed is n! 5 P 1n, n2 . It is more complicated to compute the
number of distinguishable words that can be formed with n letters when some of
the letters are duplicates.
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The formula
Section
.6 Permutations and Combinations 10-9
EXAMPLE 10 Finding the Number of Permutations of Like Things
Find the number of “words” that can be formed if each of the 6 letters of the word
little is used once.
SOLUTION For the moment, we assume that the letters of the word little are distinguishable: “LitTle.” The number of words that can be formed using each letter once is
6! 5 P 16, 62 . However, in reality we cannot tell the l’s or the t’s apart. Therefore, we
must divide by a number to get rid of these duplications. Because there are 2! orderings of the two l’s and 2! orderings of the two t’s, we divide by 2! ? 2!. The number
of words that can be formed using each letter of the word little is
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2
P 16, 62
6!
6?5?4?3?2?1
5
5
5 180
2! ? 2!
2! ? 2!
2?1?2?1
Self Check 10 How many words can be formed if each letter of the word balloon is used once? Now Try Exercise 55.
Example 10 illustrates the following general principle.
Permutations of Like Things
The number of permutations of n things with a things alike, b things alike, and
so on, is
n!
a!b! c
Self Check Answers1. ​20 2. ​60 3. ​a. ​2,520 b. ​1 4. ​296,541,907,200 5. ​24 6. ​120 7. ​252 8. ​38,220 9. ​6 10. ​1,260
Exercises 10.6
Getting Ready
You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. If E1 and E2 are two events and E1 can be done in
4 ways and E2 can be done in 6 ways, then the event
E1 followed by E2 can be done in
ways.
2. An arrangement of n objects is called a
.
3. P 1n, r2 5
4. P 1n, n2 5
5. P 1n, 02 5
6. There are
ways to arrange n things in a
circle.
7. C 1n, r2 5
8. Using combination notation, C 1n, r2 5
.
9. C 1n, n2 5
10. C 1n, 02 5
11. If a word with n letters has a of one letter, b of
another letter, and so on, the number of different
words that can be formed is
.
12. Where the order of selection is not important, we
are interested in
, not
.
Practice
Evaluate each expression.
13. P 17, 42 15. C 17, 42 17. P 15, 52 5
19. a b 4
5
21. a b 0
23. P 15, 42 ? C 15, 32 Not For Sale
14. P 18, 32 16. C 18, 32 18. P 15, 02 8
20. a b 4
5
22. a b 5
24. P 13, 22 ? C 14, 32 Not For Sale
5 4 3
25. a b a b a b 3 3 3
68
27. a b 66
Applications
Sequences,Series,and Probability
5 6 7 8
26. a b a b a b a b 5 6 7 8
100
28. a
b 99
dcb/Shutterstock.com
29. Choosing lunch ​A lunchroom has a machine with
eight kinds of sandwiches, a machine with four
kinds of soda, a machine with both white and
chocolate milk, and a machine with three kinds
of ice cream. How many different lunches can be
chosen? (Consider a lunch to be one sandwich, one
drink, and one ice cream.) 30. Manufacturing license plates ​How many six-digit
license plates can be manufactured if no license
plate number begins with 0? 31. Available phone numbers ​How many different sevendigit phone numbers can be used in one area code if
no phone number begins with 0 or 1? 32. Arranging letters ​In how many ways can the letters
of the word number be arranged? 33. Arranging letters with restrictions ​In how many
ways can the letters of the word number be arranged
if the e and r must remain next to each other? 34. Arranging letters with restrictions ​In how many
ways can the letters of the word number be arranged
if the e and r cannot be side by side? 35. Arranging letters with repetitions ​How many ways
can five Scrabble tiles bearing the letters, F, F, F, L,
and U be arranged to spell the word fluff ? 36. Arranging letters with repetitions ​How many ways
can six Scrabble tiles bearing the letters B, E, E, E,
F, and L be arranged to spell the word feeble? 37. Placing people in line ​In how many arrangements
can 8 women be placed in a line? 38. Placing people in line ​In how many arrangements
can 5 women and 5 men be placed in a line if the
women and men alternate? 39. Placing people in line ​In how many arrangements
can 5 women and 5 men be placed in a line if all the
men line up first? 40. Placing people in line ​In how many arrangements
can 5 women and 5 men be placed in a line if all the
women line up first? 41. Combination locks ​How many permutations does
a combination lock have if each combination has 3
numbers, no two numbers of the combination are
the same, and the lock dial has 30 notches? 42. Combination locks ​How many permutations does
a combination lock have if each combination has 3
numbers, no two numbers of the combination are
the same, and the lock dial has 100 notches? 43. Seating at a table ​In how many ways can 8 people
be seated at a round table? 44. Seating at a table ​In how many ways can 7 people
be seated at a round table? 45. Seating at a table ​In how many ways can 6 people
be seated at a round table if 2 of the people insist on
sitting together? 46. Seating arrangements with conditions ​In how many
ways can 6 people be seated at a round table if 2 of
the people refuse to sit together? 47. Arrangements in a circle ​In how many ways can 7
children be arranged in a circle if Sally and John
want to sit together and Martha and Peter want to
sit together? 48. Arrangements in a circle ​In how many ways can 8
children be arranged in a circle if Laura, Scott, and
Paula want to sit together? 49. Selecting candy bars ​In how many ways can 4 candy
bars be selected from 10 different candy bars? 50. Selecting birthday cards ​In how many ways can
6 birthday cards be selected from 24 different
cards? 51. Circuit wiring ​A wiring harness containing a red, a
green, a white, and a black wire must be attached to
a control panel. In how many different orders can
the wires be attached? 52. Grading homework ​A professor grades homework by randomly checking 7 of the 20 problems
assigned. In how many different ways can this be
done? 53. Forming words with distinct letters ​How many
words can be formed from the letters of the word
plastic if each letter is to be used once? 54. Forming words with distinct letters ​How many
words can be formed from the letters of the word
computer if each letter is to be used once? © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Chapter
© Istockphoto.com/NoDerog
10-10
55. Forming words with repeated letters ​How many
words can be formed from the letters of the word
banana if each letter is to be used once? 56. Forming words with repeated letters ​How many
words can be formed from the letters of the word
laptop if each letter is to be used once? 57. Manufacturing license plates ​How many license
plates can be made using two different letters followed by four different digits if the first digit cannot
be 0 and the letter O is not used? 58. Planning class schedules ​If there are 7 class periods
in a school day, and a typical student takes 5 classes,
how many different time patterns are possible for
the student? 59. Selecting golf balls ​From a bucket containing 6
red and 8 white golf balls, in how many ways can
we draw 6 golf balls of which 3 are red and 3 are
white? 60. Selecting a committee ​In how many ways can you
select a committee of 3 Republicans and 3 Democrats from a group containing 18 Democrats and 11
Republicans? 61. Selecting a committee ​In how many ways can you
select a committee of 4 Democrats and 3 Republicans from a group containing 12 Democrats and 10
Republicans? KamiGami/
Shutterstock.com
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Section
62. Drawing cards ​In how many ways can you select a
group of 5 red cards and 2 black cards from a deck
containing 10 red cards and 8 black cards? 63. Planning dinner ​In how many ways can a husband
and wife choose 2 different dinners from a menu of
17 dinners? 64. Placing people in line ​In how many ways can 7
people stand in a row if 2 of the people refuse to
stand together? 65. Geometry ​How many lines are determined by 8
points if no 3 points lie on a straight line? 66. Geometry ​How many lines are determined by 10
points if no 3 points lie on a straight line? 67. Coaching basketball ​How many different teams can
a basketball coach start if the entire squad consists
of 10 players? (Assume that a starting team has 5
players and each player can play all positions.) 68. Managing baseball ​How many different teams can
a manager start if the entire squad consists of 25
players? (Assume that a starting team has 9 players
and each player can play all positions.) .6 Permutations and Combinations 10-11
69. Selecting job applicants ​There are 30 qualified
applicants for 5 openings in the sales department.
In how many different ways can the group of 5 be
selected? 70. Sales promotions ​If a customer purchases a new
stereo system during the spring sale, he may choose
any 6 CDs from 20 classical and 30 jazz selections.
In how many ways can the customer choose 3 of
each? 71. Guessing on matching questions ​Ten words are to be
paired with the correct 10 out of 12 possible definitions. How many ways are there of guessing? 72. Guessing on true-false exams ​How many possible
ways are there of guessing on a 10-question truefalse exam, if it is known that the instructor will
have 5 true and 5 false responses? 73. Number of Wendy’s® hamburgers ​Wendy’s® Old
Fashioned Hamburgers offers eight toppings for
their single hamburger. How many different single
hamburgers can be ordered? 74. Number of ice cream sundaes ​A restaurant offers
ten toppings for their ice cream sundaes. How many
different sundaes can be ordered? Practice
Use Pascal’s Triangle to compute each combination.
75. C 18, 32 76. C 17, 42 Discovery and Writing
7 7. Prove that C 1n, n2 5 1.
78. Prove that C 1n, 02 5 1.
n
n
79. Prove that a b 5 a
b.
r
n2r
80. Show that the Binomial Theorem can be expressed
in the form
n
n
1a 1 b2 n 5 a a ban2kbk
k
k50
81. Explain how to use Pascal’s Triangle to find C 18, 52 .
82. Explain how to use Pascal’s Triangle to find C 110, 82 .
Review
Find the value of x.
83. logx 16 5 4 85. log"7 49 5 x 1
84. logp x 5 2
1
1
86. logx 5 2 2
3
Determine whether the statement is true or false.
87. log17 1 5 0 88. log5 0 5 1 b
89. logb b 5 b log7 A
A
90.
5 log7 log7 B
B
Not For Sale