Permutations

1
Permutations

• MM1D1b Calculate and use simple permutations and
  combinations
• Goal Use the formula for the number of
  permutations

2
n Factorial

• For any positive integer n, the product of the
  integers from 1 to n is called n factorial and is
  written as n!
• The value of 0! Is defined to be 1
• Ex n 4 n! 4! 1 x 2 x 3 x 4
• 4! 24

3
Permutation

• A permutation is an arrangement of objects in
  which order is important.
• The number of permutations of n objects is given
  by
• nPn n!
• The number of permutations of n objects taken r
  at a time, where r n, is given by
• nPr __n!__
• (n -
  r)!

4
Example 1 a

• Consider the number of permutations of the
  letters in the word APRIL.
• a. In how many ways can you arrange all of the
  letters?
• nPn n!
• 5P5 5!
  formula
• 5! 5 x 4 x 3 x 2 x 1
  expand
• 5! 120
  multiply

5
Example 1 b

• b. In how many ways can you arrange 3 of the
  letters? (in APRIL)
• nPr n!
• (n-r)!
• 5P3 5! formula
• (5-3)!
• 5!
• 2!
• 5 x 4 x 3 x 2! Expand 5!. Divide out
  common factorial, 2!
• 2!
• 5 x 4 x 3 Multiply
• 60

6
Example 2

• In how many ways can you arrange the letters in
  the word FLOWER?
• nPn n! formula
• 6P6 6!
• 6! 6 x 5 x 4 x 3 x 2 x 1 expand
• 720 multiply

7
Example 3

• In how many ways can you arrange 4 of the letters
  in the word PANTHER?
• nPr n!
• (n r)!
• 7P4 7!
• (7-4)!
  formula
• 7 x 6 x 5 x 4 x 3!
• 3!
  Expand 7!, divide
• 
  common factorial 3!
• 7 x 6 x 5 x 4
  multiply
• 840

8
YOU TRY!!

• Your band has written 11 songs and plans to
  record 5 of them for a CD. In how many ways can
  you arrange the songs on the CD?
• 11P5 11!
• (11-5)! formula
• 11 x 10 x 9 x 8 x 7 x 6! expand/divide
• 6!
• 11 x 10 x 9 x 8 x 7 multiply
• 55,440