Permutations

• A Permutation is an ordered arrangement of items.
• No item is used more than once.
• The order in which they occur makes a difference.
• Example. I have a pen, a die and a calculator, how many different arrangements can I have?

          P D C
  	P C D
  	D P C
  	D C P
  	C P D
  	C D P  
  
  	6
  
          We have three choices for the first item, two for the second and one for the third. 
      

• factorial is a function that takes a positive integer.
• 3! = 3×2×1 = 6
• 5! = 5×4×3×2×1 = 120
• n! = n×(n-1)×(n-2)×...×3×2×1
• 0! = 1 (by definition)
• Sometimes factorial shows up in a division problem.
• find 7!/5!

         7!   7×6×5!
        --- = ------ =   7×6 = 42
         5!    5!
      

• Permutations of n things taken rtimes is
• _nP_r = n!/(n-r)!
• Problem 41, page 572.

      n = 7
      r = 3
  
      7P3 = 7!/(7-3)! = 7!/4! = 7×6×5 = 210
      

• If I have duplicates, then the formula is n!/(p!×q!×...)
• How many different ways can you rearrange the letters in SCIENCE (Problem 50)

      n = 7
      c = 2
      e = 2
  
      7!/(2!×2!) = 7!/2×2 =  1260
      

Homework