The document introduces circular permutation as the number of ordered arrangements that can be made of n objects in a circle. It is calculated as (n-1)!. Several examples are provided to illustrate calculating the number of ways to seat people or arrange beads in a circular configuration. Groups are then asked to calculate the number of ways 4 married couples can be seated at a circular table if spouses sit opposite each other (6 ways) or if men and women alternate (144 ways).

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......