Question

(Circular Permutations) In how many ways can 7 people { A, B, C, D, E, F, G } be seated at a round table if

(a) A and B must not sit next to each other;

(b) C, D, and E must sit together (i.e., no other person can sit between any of these three)?

(c) A and B must sit together, but neither can be seated next to C or D.

Consider each of these separately.

Hint: Conceptually, think of the groups of two or three people as one "multi-person" entity in the overall circular arrangement. It may help to draw a diagram, fixing a particular person at the top of the circle (thereby eliminating the duplicates due to rotations).

Answer 1

For a round table the formulae is (n-1)!.

(1) If it would not matter if A and B sat next to each other
then the answer would be (7-1)! or 6! or 720.

But from those 6! or 720 ways, we must subtract the number of ways
those two particular persons can sit next to each other.

The cases we must subtract are the ways of seating 6 "things" around the
table.  5 single people and and one "pair".  That would be (6-1)! or 5! 
or 120 ways.

However, there are two ways the "pair"  could sit, A left of B or A right
of B.  So we double the 120 ways to 240 ways.  So we must subtract 240 ways
from the 720.

Answer: 720-240 = 480 ways.

(2) Consider C, D and E as one individual unit, now we have 4 people left, so total we have 4 + 1 ( C, D and E) = 5 units but in circular permutation, its (n - 1)! = which is 4!. And C, D and E can exchange their place with each other.

So answer would be = 4! × 3! = 24 × 6 = 144

(3) consider AB as a single unit.

Now we will find in how many ways AB can sit next to C or D. So their are 2 possible ways so do so.

1. (AB) C D E F G = 6 units, but as its circular permutation its = 5!, and A and B can exchange their places, so its 2 × 5! = 240

2. E F G C D (AB) = 6 units, but as its circular permutation its = 5! And A and B can exchange their places, so its 2 × 5! = 240

Total ways = 240 + 240 = 480

So total possible ways of A and B together but not next to C or D = 720 - 480 = 240

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