The document discusses circular permutations, which refers to the number of arrangements that can be made of n objects arranged in a circle. It gives the formula for calculating circular permutations as (n-1)!. It provides examples of using this formula to calculate the number of ways to seat people or arrange beads in a circular configuration. It also discusses a special case where the objects can be flipped over, in which case the formula is (n-1)!/2. Finally, it asks how many ways 4 married couples can be seated at a circular table if spouses sit opposite each other or if men and women alternate.

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!
B B
C
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!
en
1 c
f m
ho 3! x 4!
ice o
s
of
ice
lady 1 6 x 24
m o
ch
an
4
144
ways to seat a men and a
lady 2
lady 4 women alternate on a circular
table
en 3 c
f m ho
ice
s o lady 3 s o
ce
oi f m
ch en
2

8. The next scribe is Mary
Ann......