Introduction to Permutations

• A permutation is an arrangement of objects in a specific order.
• The order of arrangement is crucial in permutations.
• Example: Arranging the letters of the word “ABC”.
• Total permutations = $3! = 6$.

Key Formula for Permutations

• The number of permutations of \(n\) objects taken \(r\) at a time is given by:
  \[
  P(n, r) = \frac{n!}{(n-r)!}
  \]
• Here, \(n!\) (factorial) represents the product of all integers from 1 to \(n\).
• Example: Find the number of ways to arrange 3 objects out of 5:
  \[
  P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60
  \]

Derivation of the Formula

• Consider \(r\) vacant positions to fill using \(n\) objects.
• The first position can be filled in \(n\) ways.
• The second position can be filled in \((n-1)\) ways.
• Continue this process until \(r\) positions are filled.
• Total permutations:
  \[
  P(n, r) = n \times (n-1) \times \ldots \times (n-r+1)
  \]
• Simplify using factorials:
  \[
  P(n, r) = \frac{n!}{(n-r)!}
  \]

Example Problems

• Example 1: How many 4-letter words can be formed from the letters of “ROSE” without repetition?
  \[
  P(4, 4) = \frac{4!}{(4-4)!} = 4! = 24
  \]
• Example 2: How many ways can a committee of 2 be formed from 6 people?
  \[
  P(6, 2) = \frac{6!}{(6-2)!} = \frac{720}{24} = 30
  \]

Questions to Consider

• How does the formula change if repetition is allowed?
• What happens when \(r = n\)?
• How does permutation differ from combination?

PDF Presentation

Video

Simulation

Permutation Quizzes

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