CIRCULAR PERMUTATIONS
Circular permutations are mostly involving round table seating problems or rectangular table
seating problems.
1) n people seating around a round table. We will get different solution with regard to two
different cases, that are orientation matters or not.
If orientation matters the total number of arrangements is
If orientation does not matter the total number of arrangements is
, that shows the n people seating at round table if they rotate one nth of a circle the
seating pattern will remain the same pattern.
For example, two people, A and B, seating at a round table, if orientation matters, say
whoever sits facing south is more important, there are 2 ways of seating. If orientation
does not matter, there is only one way of seating. If three people, A, B, and C are sitting at
a round table with orientation of concern, there are 6 ways of seating as shown below
A
B
A
C
C
(1)
B
B
(2)
A
B
C
(3)
C
C
A
(4)
A
C
B
(5)
B
A
(6)
If orientation does not matter, (1), (4), and (5) will be considered exactly the same and so
do (2), (3), and (6). There are only 3!/3 = 2 ways of seating.
Summary:
Where:
is the number of arrangements if orientation does not matter,
is the number of arrangements if orientation matters,
n
2.
is the number of arrangements for the first element.
Problems of Couples Sitting at a Round Table
Added restrictions to sitting problems would make the solution of the problems with more
difficulties. Couples sitting problems are good examples.
a) Two couples sitting at a round table
a1) The husband sits opposite his wife for each couple
Case 1:
Orientation matters
Solution 1:
two elements permutation with each couple flips over
H1
H2
X
W2
H1
W2
X
H2
H1
H2
#
H2
W1
W1
H1
W1
W1
W2
W2
X W1
W1
W2
X W2
H2
W2
H1
Solution 2:
W2
H2
H1
H1
#
H2
W1
W1
H1
H2
The first couple may sit in 4 different ways and after they have sit down the
second couple will have 2 choices left for them.
4x2 = 8
Case 2:
Orientation does not matter
2 is the solution because the first couple may sit any ways and the second will
have two ways to sit down.
These four arrangements with an X will become the same and the remaining four
will become another one. If orientation does not matter the first couple sitting
in any of the four positions will become only one arrangement, therefore , the
solution is