# In how many ways can six people be seated in a circle with ten seats?

I'm having troubles with solving this question:

We have 6 people. In how many ways we can put them in a line?

-The answer is $720$, because $6!=720$.

We want to put them in circle with 6 seats.

-The answer is $5!=120$.

In circle with $10$ seats?

-The answer is $15120$, but I can't figure it out why is that so.

• @kimiTanaka The OP understands that. See his answer to the second question. The issue is how to handle the four empty seats. – N. F. Taussig Oct 1 '17 at 14:26
• @N.F.Taussig Thanks. – kimi Tanaka Oct 1 '17 at 14:32

Note that in circular arrangements, there is no sense of first and last. But if you put one person on any seat on circular table, he can be used as a marker of beginning/end. Thus this circular arrangement can be treated as a linear arrangement now!

For second problem: We put a person on any seat. Now only 5 people are left to put on 5 seats. This can be done in $5!$ ways.

For the third problem: If you want to put in circle with 10 seats, then first put 1 person on any seat. Now, you have 5 persons to put on 9 seats, So the answer is $\binom{9}{5} \cdot 5!$

• Thank you for explanation, but i still don't understand why is there 9C5 – Mirjan Pecenko Oct 1 '17 at 14:32
• You have put $1$ person at a seat. Call him $P$. Now we can think of this as now this has converted into linear arrangement where $P$ is always at front, ie his position is fixed. You have $9$ seats and $5$ persons remaining. Choose $5$ seats from $9$, and then arrange persons among themselves. – jonsno Oct 1 '17 at 14:37
• Oh, nice.. Than you very much, now I clearly understand the problem. – Mirjan Pecenko Oct 1 '17 at 14:42

In the circle there is no way to tell who is first. Put someone in the circle and then fill in the remaining 5. How we fill in those 5 is what matters.

For the circle with 10 spots, think of it like the circle with 6 spots, but now with 4 indistinguishable people. That is, there is no difference between one empty seat and another. The only difference comes from the people around them.