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Section 10.1
Objective: Use the fundamental counting principle and permutations (D.4.B)
Example 1: A sporting goods store offers 3 types of snowboards (all­mountain,
freestyle, and carving) and 2 types of boots (soft and hybrid). How many
choices does the store offer for snowboarding equipment?
Solution: Draw a tree diagram
Try:
Applebee’s is running a special ‘You Pick 3’ meal deal. You have your choice
of one of three appetizers (mozzarella sticks, chicken fingers, or chips and
salsa), 3 entrée’s (pasta, sirloin steak, or chicken casear salad), and 2 desserts
(ice cream brownie sundae or apple crisp). Make a tree diagram to determine
how many meal choices you have.
How could we arrive at this same answer without having to create a tree
diagram?
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Fundamental Counting Principle Two events: If one event can occur in m ways and another event can occur in n
ways, then the number of ways that both events can occur is _m x n_.
Three or more events: The fundamental counting principle can be extended to
three or more events. For example, if three events can occur in m, n, and p
ways, then the number of ways that all three events can occur is _m x n x p .
Example 2
A baseball manager has five possible starting pitchers for a game. He also must
decide which of three catchers and four first basemen to put in the starting
lineup. How many ways can he choose the players for these positions?
Example 3
The standard configuration for a Texas license plate is 1 letter followed by 2
digits followed by 3 letters. How many different license plates are possible if:
a. Digits and letters are repeated
b. Digits and letters aren't repeated
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More Examples:
Determine the number of Washington License Plates possible if a plate must contain 3 letters followed by 4 digits and:
a) Letters and Digits can be repeated
b) Only digits can be repeated 3
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An ordering of n objects is a permutation of the objects. For instance, there are 6 permutations of the letters A, B, and C: ABC ACB BAC BCA CAB CBA
3 letters for the first choice, 2 letters for the second choice, and one letter for the third choice
How could we arrive at the number 6 without having to generate a list of the
possible orders?
Try:
There are 5 people waiting in line at Subway. How many ways can they
place their order?
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Example 4
Ten teams are competing in the final round of the Olympic four­ person
bobsledding competition
a. In how many different ways can the bobsledding teams finish the
competition? (Assume there are no ties.)
b. In how many different ways can 3 of the bobsledding teams finish first,
second, and third to win the gold, silver, and bronze medals?
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The answer in part b is called a ‘permutation of 10 objects taken 3 at a time’
and is denoted 10P3. We can write a special formula evaluate this expression.
Permutations of n Objects Taken r at a Time The number of permutations of r objects taken from a group of n distinct objects
is denoted nPr and is given by this formula:
P =
n r
n! is read as ‘n factorial’ and is equivalent to n x(n­1)x(n­2)
x...x 3 x 2 x 1. The number of permutations of n objects is n!. 6
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Example 5
How many different ways can 4 raffle tickets be selected from 50 tickets if each
ticket wins a different prize?
Try:
You have 6 homework assignments to complete over the weekend. However, you only
have time to complete 4 of them on Saturday. In how many orders can you complete 4
of the assignments?
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Permutations with Repetition – Consider the letters S, O, S. If you consider S
and S to be distinct, then there are 6 permutations,
SOS SSO OSS OSS SOS SSO
But, if the two occurrences of S are considered to be interchangeable, then
there are only _____ distinguishable permutations:
Permutations with Repetition
The number of distinguishable permutations of n objects where one object is
repeated s1 times, another is repeated s2 times, and so on is:
n!/(s1!s2!...sk!)
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Example 6
Find the number of distinguishable permutations of the letters in
a. SOCCER
b. SWIMMING
Try:
Find the number of distinguishable permutations of the letters in:
MISSISSIPPI
OFFENBURGER
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