Sullivan Algebra and Trigonometry: Section 14.2

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Sullivan Algebra and Trigonometry Section 14.2

• Objectives of this Section
• Solve Counting Problems Using the Multiplication
  Principle
• Solve Counting Problems Using Permutations
• Solve Counting Problems Using Combinations
• Solve Counting Problems Using Permutations
  Involving n Non-Distinct Objects

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Multiplication Principle of Counting
If a task consists of a sequence of choices in
which there are p selections for the first
choice, q selections for the second choice, r
selections for the third choice, and so on, then
the task of making these selections can be done
in
different ways.
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If a license plate consists of a letter, then 5
numbers, how many different types of license
plates are possible?
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A permutation is an ordered arrangement of n
distinct objects without repetitions. The symbol
P(n, r) represents the number of permutations
of n distinct objects, taken r at a time, where r
lt n.
Example How many ways can you arrange 7 books on
a shelf?
There are 7 choices for the first book, six
choices for the second book (since the first was
already picked), etc...
So (7)(6)(5)(4)(3)(2)(1) 7! 5040 ways
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Number of Permutations of n Distinct
Objects Taken r at a Time
The number of different arrangements from
selecting r objects from a set of n objects
(r lt n), in which
1. the n objects are distinct
2. once an object is used, it cannot be repeated
3. order is important
is given by the formula
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Suppose you were in charge of selecting four
performers from a group of twelve to perform at a
talent show. How many ordered arrangements of
performers do you have to choose from?
Since you are selecting 4 from 12, find P(12,4)
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A combination is an arrangement, without regard
to order, of n distinct objects without
repetitions. The symbol C(n, r) represents the
number of combinations of n distinct objects
taken r at a time, where r lt n.
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Number of Combinations of n Distinct Objects
Taken r at a Time
The number of different arrangements from
selecting r objects from a set of n objects
(r lt n), in which
1. the n objects are distinct
2. once an object is used, it cannot be repeated
3. order is not important
is given by the formula
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How many ways can you form a committee of three
people from a group of 25?
Since you are selecting 3 from 25 and order does
not matter, we find