Permutations

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Permutations
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Basic Counting Principal

• If there are m ways to make a first selection and
  n ways to make a second selection, there are m x
  n ways to make the two selections.

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Example 1

• Suppose there are two ways to get from Austin to
  Dallas, and four ways to get from Dallas to
  Tulsa, OK. How many routes are there from Austin
  to Tulsa through Dallas?
• 2 x 4 8
• 8 ways to get from Austin to Tulsa through Dallas.

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Example 2

• The school cafeteria offers four kinds of bread,
  five kinds of meat, and three kinds of cheese on
  their sandwiches. How many unique sandwiches are
  offered by the cafeteria?

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Permutations

• A common kind of counting problem is to find the
  number of arrangements of a set of objects.
• Each arrangement is called a permutation.
• A permutation is an arrangement of some or all of
  a set of objects in a specific order.

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Example 3

• A bag contains 26 alphabet blocks, one for each
  letter of the alphabet. If you pick out 4 blocks
  at random, how many possible letter combinations
  are possible?
• How many will we choose?
• ____ x ____ x ____ x ____
• How many to choose from for the first blank?

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Example 3 (cont)

• 26 x ____ x ____ x ____
• Now, how many letters to choose from for the
  second space?
• 26 x 25 x ____ x ____
• Continue for the next two spaces
• 26 x 25 x 24 x 23
• 358,800

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Example 3 (cont)

• This would be written as 26P4
• It is read as the permutation of 26 things taken
  4 at a time.

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Formula

• n is the number of objects from which to
  choose.
• r is the number of objects you will choose.

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Example 4

• There are 24 students in homeroom. One will read
  announcements, one will take attendance, and one
  will pass out notices. In how many ways can
  three student helpers be chosen to do these
  tasks?
• 24P3