The Counting Principle and Permutations

• This is 11.7
• Remember: The Counting Principle states that if a first experiment can be performed m ways and a second can be performed n ways then the two experiments in that order can be performed m·n ways.
• If we want to count the number of password we can form with three letters.
  • Determine with replacement or without.
  • Draw three blanks
  • How many can go in the first blank?
  • How many in the second ?
  • How many in the third?
  • Multiply these, this is the answer.
  • This is a direct application of the counting principle.
• Notation: Factorial
  • 0! = 1
  • n! = n*(n-1)·(n-1) ··· 3·2·1 for n > 0;
  • Or n! = n·(n-1)! for n > 0
• You should have a factorial button on your calculator.
  • Warning: Factorial gets large quickly.
• A permutation is any ordered arrangement of a given set of objects.
• By order we mean that the order is important, does the thing selected first matter.
  • We hold a random drawing for three people to go to lunch for free
  • Does the order matter?
  • The first gets to go to McDonalds, the second to Flip, and the third to the The Dinor.
  • Does the order matter now?
  • In general the order matters if there is a difference in the positions for some reason.
• The number of permutations possible when r objects are selected from n objects is
  • $_nP_r=\frac{n!}{(n-r)!}$
• But you should have an $_nP_r$ button on your calculator.
• The development of this is not that hard, but we will skip it.
• We use permutations when order matters and we are selecting without replacement
• Try some problems on 695
• Try some problems on 696-697