Question

7 children arrange in a circle having 10 identical seats. Total arrangements possible?

Answer 1

Since they are seated in a circle, each seat will be similar to start from. Which means that the first seat will always be constant. After the first seat is taken, we will then begin to have different arrangement of seatings for each person.

So the first person will pick a seat. Then the second person will pick one of 9 seats. The third will pick one of 8 seats, and so on.

Therefore, the way we choose is 9C7 = 36 unordered ways, note that since the first seat is always taken, our total is 9 not 10 (circle arrangement is different than a row).

There are 7! arrangements for the people.

Therefore, the solution is 7!*36 = 181,440 ordered ways.