# Ways to seat r out of n people around a table

• changeofplans
In summary, the formula for the number of ways to seat r of n people around a circular table, where seatings are considered the same if every person has the same two neighbors, is \frac{P(n,r)}{2r}, where P(n,r) is the number of permutations of r people from a group of n people. This accounts for the possibility of shifting the seating arrangement by one seat and the mirrored positions.
changeofplans

## Homework Statement

Find a formula for the number of ways to seat r of n people around a circular table, where seatings are considered the same if every person has the same two neighbors without regard to which side these neighbors are sitting on.

None

## The Attempt at a Solution

I understand the more basic form of this question; in order to seat n people, we have $\frac{n!}{n}$, since there are n! ways to seat n people and we divide out the n repetitions.

From there, we see that the ways to seat r of n people is simply $\frac{P(n, r)}{r}$. So that gives us the first part of the problem.

But I'm not really sure where to go from here. I'm thinking, in this specific problem, we shouldn't be dividing out the r people, since it's explicitly stated that the only seating arrangements that are considered the same are where the same person has the same two neighbors. So I'm thinking, I definitely have the numerator: P(n, r). And.. then I hit a wall.

Any help is appreciated.

You'd still have to divide by r because if you shift everyone by one seat, they'll still have the same two neighbors. There's still another way to rearrange a permutation so that everyone has the same two neighbors. You need to account for that.

Okay, that makes sense. So now I have:

$\frac{P(n,r)}{r}$

What about multiplying r by 2, for the mirrored positions?

$\frac{P(n,r)}{2r}$ = $\frac{n!}{\frac{(n-r)!}{2r}}$

Sounds good.

## 1. How many ways can r people be seated around a table?

The number of ways r people can be seated around a table is determined by the formula nPr = n! / (n-r)!, where n is the total number of people and r is the number of people to be seated. For example, if there are 5 people and we want to seat 3 of them, the number of ways would be 5P3 = 5! / (5-3)! = 5! / 2! = 5*4*3 = 60 ways.

## 2. Does the seating arrangement matter?

Yes, the seating arrangement does matter. In a circular table, the arrangement of people can change the dynamics of a conversation. In a rectangular or square table, the seating arrangement can also affect the interactions between individuals.

## 3. Can two people sit in the same seat?

No, two people cannot sit in the same seat. This is because for r people to be seated around a table, we are assuming each person occupies one seat only. If two people were to sit in the same seat, the total number of people seated would be more than r, and the formula nPr = n! / (n-r)! would not be valid.

## 4. How does the number of people affect the total number of seating arrangements?

The more people there are, the greater the number of seating arrangements. This is because there are more options for people to be seated in different positions. For example, if there are 8 people and we want to seat 4 of them, the number of ways would be 8P4 = 8! / (8-4)! = 8! / 4! = 8*7*6*5 = 1680 ways.

## 5. Can the order of seating be changed?

Yes, the order of seating can be changed. This is because the formula nPr = n! / (n-r)! takes into account the different ways in which r people can be arranged. For example, if there are 6 people and we want to seat 3 of them, the number of ways would be 6P3 = 6! / (6-3)! = 6! / 3! = 6*5*4 = 120 ways. This means that the order of the 3 people sitting around the table can be changed in 120 different ways.

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