2.1 Factorial Notation

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2.1 Factorial Notation

• (Textbook Section 4.6)

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Warm Up Question

• How many four-digit numbers can be made using the
  numbers 1, 2, 3, 4?
• (all numbers must only be used once for each
  4-digit number)

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Fundamental Principle of Counting

• If one operation can be done in m ways and
  another operation can be done in n ways, then
  together, they can be done in mxn ways
• This principle can be extended to any number of
  operations
• i.e. if operation A can be done 3 ways, operation
  B can be done 4 ways, operation C can be done in
  2 ways and operation D can be done 7 ways, then
  together they can be done in 3 x 4 x 2 x 7 168
  ways

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Back to Warm Up Question

• How many ways can we place the number 1?
• 1 can be the 1st, 2nd, 3rd or 4th digit, so 4
  ways
• IF we have placed the number 1, how many ways can
  we place the number 2?
• One of the digits has already been taken up by
  the number 1, so there are 3 remaining digit
  places to put the number 2, so 3 ways

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Back to Warm Up Question (Continued)

• IF we have placed numbers 1 and 2, how many ways
  can we place the number 3?
• 2 remaining digit places, so 2 ways
• IF we have placed numbers 1, 2, and 3, how many
  ways can we place the number 4?
• One spot remaining, place 1 there, so 1 choice
• How many ways to place all 4 digits?
• 4 x 3 x 2 x 1 24 ways

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Factorial Notation

• Many counting and probability calculations
  involve the product of a series of consecutive
  integers (i.e. 4x3x2x1)
• We can write these products using Factorial
  Notation
• The symbol for this notation is
• n! or x!

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How to Use Factorial Notation

• For all natural numbers (integers gt 0) n!
  represents the product of all natural numbers
  less than or equal to n
• n! n x (n-1) x (n-2) x 3 x 2 x 1
• i.e. 5! 5 x 4 x 3 x 2 x 1 120

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Rules for Factorial Notation

• 0! 1
• n!/n! 1
• n!/0! n!/1 n!