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Circular Permutation

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  • 1. CIRCULAR PERMUTATION
  • 2. Hello folks this is Precious your scribe for today.  Today Mr. K introduced to us the Circular Permutation.  What is Circular  permutation? Circular Permutation is the number of ordered arrangements that can be  made of n objects in a circle is given by:                                                                                                 ( n ‐ 1 ) ! and in special problems like bracelets and  necklaces that can flip over we can use:                                              ( n ‐ 1 ) ! 2
  • 3. Example number 1: How many distinguishable ways can 3 people be seated around a  circular table? hint: **person 3 is our point of reference Solution: ( n ­ 1 ) ! ( 3 ­ 1 ) !     2! person 3  2 x 1 person 2 person 1     2 person 3 person 1 person 2 therefore there are 2 ways to seat 3 people in a circular table.
  • 4. Example number 2: How many distinguishable ways can 4 people be seated around a circular  table? hint: quot;Aquot; is our point of reference Solution: A A A ( n ­ 1)! B D D C C D ( 4 ­ 1)!     3! C B B 3 x 2 x 1 A A A      6 B D B C B C threrefore there are 6  ways to seat 4 people  C D D in a circular table
  • 5. Example number 3: How many distinguishable ways can 4 beads be arranged on a circular bracelet? Solution: ( n ­ 1)! 2 bead 1 ( 4 ­ 1 )! bead 1 2 3! bead 4 bead 2 bead 3 2 bead 4 3 x 2 x 1 bead 3 bead 2 2 6 bead 1 2 3 bead 2 bead 3 hint: bead 1 is our point of  bead 4 reference
  • 6. And then Mr. K decides to form us into groups to solve this problem: In how many ways can 4 married couples seat themselves around a  circular table if: a.) spouses sit opposite each other? Solution: ( n ­ 1 )! **here we have our formula then we  ( 4 ­ 1 )! know that there is 4 spouses subtract 1       3! and then factorial. 3 x 2 x 1 6
  • 7. b.) men and women alternate? Solution: ladies         x         men ( 4 ­ 1 )!       x        4! 3!              x            4! lady 1 6              x            24 4 choices of men 1 choice of man 144 ways to seat a men and a  lady 2 lady 4 women alternate on a circular  table 2 choices of men 3 choices of men lady 3
  • 8. The next scribe is Mary  Ann......