# Circular permutation - Arranging 4 persons around a circular table where 8 seats are there.

Suppose 4 persons $A,B,C$ and $D$ sit around a round table with 8 seats. Rotation by 8,16,24,... seats defines same arrangement and other rotations gives different arrangements. Find the number of ways that these four people can be seated at the round table.

My solution:

Place one person in any seat; that is a reference seat.

Now 3 persons in 7 seats gives $7\times6\times5$ arrangements (if seats are not labeled)

or $8\times7\times6\times5$ (if seats are labeled)

Is this approach right?

• Your answers are correct. – N. F. Taussig Mar 12 '16 at 14:44

## 1 Answer

simply thinking your way. The chair can be anyone if the $8$ so its ${8\choose 1}$ so it should be $8.7.6.5=1680$ alternatively they are simple permutations so its just $8P4$ which yields the same answer.

• I guess your answer 1680 assumes that char is distinct. If chairs are identical(unmarked), then it will be 7*6*5 right? – Kiran Mar 12 '16 at 13:49
• because, if the chair is identical, due to rotational symmetry, it must be 1680/8 = 210, correct? – Kiran Mar 12 '16 at 13:50
• Oh yes . you havent mentioned it so i assumed them to be distinct – Archis Welankar Mar 12 '16 at 13:52