Circular arrangements

Turning the table

Five friends, Amit, Bela, Chen, Dara and Esha, sit down to dinner at a round table. In how many ways can they sit?

In a row, the answer would be $5! = 120$. Round a table, something changes. Suppose everyone stands up and moves one seat clockwise. Each person still has the same friend on the left and the same friend on the right. Nobody can tell that anything happened. So we count it as the same seating.

Every round seating can be read as a row in 5 ways, one for each seat we start from. So the 120 rows come in groups of 5, and each group is one round seating:

For $n$ people the same idea gives

Another way to see it. Seat Amit anywhere. Since the table can turn, his seat does not matter. Now read the other seats clockwise from Amit. They form a row of $n - 1$ seats, filled in $(n - 1)!$ ways.

When the seats are different

The rule $(n - 1)!$ needs seats that look alike. If the seats are numbered, or one seat is beside a window, then turning the table changes the seating. Then the people fill $n$ different seats in $n!$ ways, exactly as in a row. Read every question for this.

Necklaces and garlands

A necklace is different from a table in one more way. You can pick it up and turn it over. Then the clockwise order A, B, C, D, E reads A, E, D, C, B. It is the same necklace.

So each necklace is counted twice among the $(n - 1)!$ round orders, once for each side. For $n$ beads, all different and $n \ge 3$:

Example 1. Seven beads of different colours make $\dfrac{6!}{2} = 360$ necklaces.

People at a table are not halved. A person can tell a friend on the left from a friend on the right, so the clockwise and anticlockwise seatings are different.

Set $n = 3$. Three people sit round a table in 2 ways, but only 1 necklace can be made from 3 beads. Turning it over joins the two orders.

Conditions at a round table

The tricks of rows still work. Seat the units, then sort out the inside of each unit.

Example 2 (together). Eight guests sit at a round table, and Rhea and Tomas must sit next to each other.

1. Tie them together. Now there are 7 units round the table: $6! = 720$ ways.
2. The pair inside. Rhea and Tomas can swap: $2! = 2$ ways.
3. Multiply. $720 \times 2 = 1440$ seatings.

Example 3 (apart). The same eight guests, but now Rhea and Tomas must not sit side by side. Count all the seatings and take away the ones with the pair together:

Example 4 (alternate). Four boys and four girls sit at a round table, boys and girls taking turns.

1. Seat the boys first. Round the table: $3! = 6$ ways.
2. The girls. The boys leave 4 gaps, one between each pair of neighbours. The girls fill them in $4! = 24$ ways. The seats are now different, since the boys mark them.
3. Multiply. $6 \times 24 = 144$.

Practise with new numbers

Try it in Python

The program lists every order of the people, turns each one so that A comes first, and counts what is left. Turning over is checked too, for the necklaces.

Common mistakes

• Using $n!$ for a round table. Turning the table gives the same seating, so divide by $n$.
• Halving for people. Only a necklace or a garland can be turned over. People at a table are not halved.
• Forgetting the order inside a pair. Two people tied together can swap, so multiply by $2!$.
• Numbered seats. If the seats are different, the count is $n!$, as in a row.

Remember these

• $n$ people round a table: $(n - 1)!$ ways.
• $n$ different beads on a necklace or a garland: $\dfrac{(n - 1)!}{2}$, for $n \ge 3$.
• Fix one person first, then fill the seats from there.
• "Together": tie them, then multiply by their own orders. "Apart": all minus together.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In how many ways can 9 friends sit at a round table?

2. Neha says 6 people can sit at a round table in $6! = 720$ ways. When is she right, and what is the answer otherwise?

3. How many garlands can be made from 10 flowers, all different?

4. Three married couples sit at a round table, men and women taking turns. In how many ways can they sit?