Transcript 10.6

Chapter 10
Sequences, Induction,
and Probability
10.6 Counting Principles,
Permutations, and
Combinations
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1
Objectives:
•
•
•
•
Use the Fundamental Counting Principle.
Use the permutations formula.
Distinguish between permutation problems and
combination problems.
Use the combinations formula.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
2
The Fundamental Counting Principle
The number of ways in which a series of successive
things can occur is found by multiplying the number of
ways in which each thing can occur.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
3
Example: Using the Fundamental Counting Principle
A pizza can be ordered with three choices of size (small,
medium, or large), four choices of crust (thin, thick, crispy,
or regular), and six choices of toppings (ground beef,
sausage, pepperoni, bacon, mushrooms, or onions). How
many different one-topping pizzas can be ordered?
We use the Fundamental Counting Principle to find the
number of different one-topping pizzas that can be ordered.
Size – 3 choices
3 4 6  72
Crust – 4 choices
Toppings – 6 choices
72 different one-topping
pizzas can be ordered.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
4
Permutations
A permutation is an ordered arrangement of items that
occurs when
No item is used more than once.
The order of arrangement makes a difference.
The number of possible permutations if r items are
taken from n items is
n!
n Pr 
(n  r )!
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
5
Example: Using the Formula for Permutations
In how many ways can six books be lined up along a
shelf?
Because we are using all six books in every possible
arrangement, we are arranging r = 6 books from a group
of n = 6 books.
n!
6!
6! 6!
P


P


  720
n r
6 6
(n  r )!
(6  6)! 0!
1
Six books can be lined up along a shelf in 720 ways.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
6
Combinations
A combination of items occurs when
The items are selected from the same group.
No item is used more than once.
The order of the items makes no difference.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
7
Example: Distinguishing between Permutations and
Combinations
For each of the following problems, explain if the problem
is one involving permutations or combinations.
a. How many ways can you select 6 free DVD’s from a list
of 200 DVD’s?
Because order makes no difference, this is a combination.
b. In a race in which there are 50 runners and no ties, in
how many ways can the first three finishers come in?
The order in which each runner finishes makes a
difference, this is a permutation.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
8
Combinations of n Things Taken r at a Time
The number of possible combinations if r items are
taken from n items is
n!
n Cr 
(n  r )!r !
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
9
Example: Using the Formula for Combinations
From a group of 10 physicians, in how many ways can
four people be selected to attend a conference on
acupuncture?
We are selecting r = 4 people from a group of n = 10
people.
n!
10!
10! 10 9 8 7 6!
C




 210
n r
(n  r )!r ! (10  4)!4! 6!4!
6! 4 3 2 1
Four people can be selected from a group of 10 in 210 ways.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
10