Question

A circular table has 6 chairs, five of which are identical. In how many ways can six people be arranged on these chairs?​

Answer 1

Answer:

If the question was to find the number of ways of making six people sit on six chairs, the answer would have been 6! which is equal to 6x5x4x3x2x1=720.

However, in the case of a circular table, we assume all the seats are similar and the only thing matters is the order of sitting, i.e; who is sitting next to whom and not who is sitting where.

Lets name the seats around the table A,B,C..F. So assume I am placing the first person somewhere around the table (at A, say) and that becomes our reference point. Now our aim is to arrange the remaining 5 persons on the seats from B to F.

Since seat A can be filled only by one person ( first person, our reference), seat B can be occupied by any of the remaining 5 ( anyone but first person who already occupied A), seat C can be occupied by any of the remaining 4 and so on..

So the number of ways 6 people can be seated around a round table becomes 1x5x4x3x2x1=120.

In other words it is equal to (n-1)!

Step-by-step explanation: