Transcript Mathematical Ideas - Millersville University of Pennsylvania

Chapter 11
Counting
Methods
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 11: Counting Methods
11.1
11.2
11.3
11.4
11.5
Counting by Systematic Listing
Using the Fundamental Counting Principle
Using Permutations and Combinations
Using Pascal’s Triangle
Counting Problems Involving “Not” and
“Or”
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Chapter 1
Section 11-3
Using Permutations and Combinations
11-3-3
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Using Permutations and
Combinations
• Permutations
• Combinations
• Guidelines on Which Method to Use
11-3-4
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Permutations
In the context of counting problems,
arrangements are often called permutations;
the number of permutations of n things taken r
at a time is denoted nPr. Applying the
fundamental counting principle to
arrangements of this type gives
nPr =
n(n – 1)(n – 2)…[n – (r – 1)].
11-3-5
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Factorial Formula for Permutations
The number of permutations, or
arrangements, of n distinct things taken r at a
time, where r  n, can be calculated as
n!
.
n Pr 
(n  r )!
11-3-6
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Example: Permutations
Evaluate each permutation.
a) 5P3
b) 6P6
Solution
5!
5!
a) 5 P3 
  60
(5  3)! 2!
6!
6!
b) 6 P6 
  720
(6  6)! 0!
11-3-7
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Example: IDs
How many ways can you select two letters followed
by three digits for an ID if repeats are not allowed?
Solution
There are two parts:
1. Determine the set of two letters.
2. Determine the set of three digits.
26 P2  10 P3
 650  720  468,000
Part 1 Part 2
11-3-8
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Example: Building Numbers From a Set
of Digits
How many four-digit numbers can be written using
the numbers from the set {1, 3, 5, 7, 9} if repetitions
are not allowed?
Solution
Repetitions are not allowed and order is important, so
we use permutations:
5!
5!
  120.
5 P4 
(5  4)! 1!
11-3-9
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Combinations
In the context of counting problems, subsets,
where order of elements makes no difference,
are often called combinations; the number of
combinations of n things taken r at a time is
denoted nCr.
11-3-10
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Factorial Formula for Combinations
The number of combinations, or subsets, of n
distinct things taken r at a time, where r  n,
can be calculated as
n Pr
n!

.
n Cr 
r ! r !(n  r )!
Note: n Cr 
n!
n!

n Cnr .
r !(n  r )! (n  r )!r !
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11-3-11
Example: Combinations
Evaluate each combination.
a) 5C3
b) 6C6
Solution
5!
5!
a) 5C3 

 10
3!(5  3)! 3!2!
6!
6!
b) 6C6 

1
6!(6  6)! 6!0!
11-3-12
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Example: Finding the Number of Subsets
Find the number of different subsets of size 3
in the set {m, a, t, h, r, o, c, k, s}.
Solution
A subset of size 3 must have 3 distinct elements,
so repetitions are not allowed. Order is not
important.
9!
9!

 84
9 C3 
3!(9  3)! 3!6!
11-3-13
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Example: Finding the Number of Poker
Hands
A common form of poker involves hands (sets) of five
cards each, dealt from a deck consisting of 52 different
cards. How many different 5-card hands are possible?
Solution
Repetitions are not allowed and order is not
important.
52!
52!

 2,598,960
52 C5 
5!(52  5)! 5!47!
11-3-14
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Guidelines on Which Method to Use
Permutations
Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important.
Order is not important.
Arrangements of n items
taken r at a time
nPr = n!/(n – r)!
Subsets of n items taken r
at a time
nCr = n!/[ r!(n – r)!]
Clue words: arrangement, Clue words: group,
schedule, order
sample, selection
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11-3-15