Circular Permutation

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  • This regards slide 6
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  • I think this is incorrect.
    First, form a dashed octagon so that you can insert the ways to seat people. Now:
    There are 8 ways to seat the first person. So place an 8 at any spot Directly opposite that person, the spouse must sit. So there is only one way to seat that person. Place a 1 directly opposite the 8.
    Now that you have seated one couple, there are obviously 6 people people left. Thus the next person can be seated 6 ways. For simplicity, place a 6 directly left of the 8. Opposite the 6, the spouse must sit. So again, place a 1. Now that two more people have been seated, there are 4 ways to seat the next spot at the table. Again, for simplicity, place a 4 directly to the left of the 6. Opposite the 4, place a 1, because the spouse is the only person who can sit opposite the 4. 6 people have now been seated. Thus two people remain and there are 2 ways to seat the next person. Place a 2 directly to the left of the 4. Finally, place the last 1 directly across from the 2.

    Thus, moving counter-clockwise, and starting from the top of the circle, your numbers would be as follow:
    8,6,4,2,1,1,1,1
    Thus there are 8X6X4X2X1X1X1X1 ways to seat these people = 384 ways

    Just shifting any couple around the table gives you 7 ways to seat that couple, which is already above the ways listed.
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Circular Permutation

  1. 1. CIRCULAR PERMUTATION
  2. 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. 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. 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. 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. 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. 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. 8. The next scribe is Mary  Ann......

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