Seat arrangements around a rectangular table

[songoku]

Suppose 6 people want to sit in circular table. This case is considered as cyclic permutation.

How about if the 6 people want to sit around a rectangular table?

1.
Let say they sit on the longer sides of the table, 3 people on each side and no one sit on shorter side of the table. Is this cyclic permutation?

2.
Let say A sits on the shorter side of the table and B sits on the other shorter side, and the rest 4 people sit on the longer side, 2 people on each of the longer side. Is this cyclic?

3.
Let say now there are 8 people. Six people sit on longer side, 3 people on each side and two people sit on shorter side, 1 person on each of the shorter side. Is this cyclic?

Thanks

 

[pbuk]

Look up the definition of a cyclic permutation. What does your first statement about a circular table mean?

 

[tnich]

Suppose 6 people want to sit in circular table. This case is considered as cyclic permutation.

How about if the 6 people want to sit around a rectangular table?

1.
Let say they sit on the longer sides of the table, 3 people on each side and no one sit on shorter side of the table. Is this cyclic permutation?

2.
Let say A sits on the shorter side of the table and B sits on the other shorter side, and the rest 4 people sit on the longer side, 2 people on each of the longer side. Is this cyclic?

3.
Let say now there are 8 people. Six people sit on longer side, 3 people on each side and two people sit on shorter side, 1 person on each of the shorter side. Is this cyclic?

Thanks

Why should the shape of the table make a difference? The table has a continuous edge. Putting corners in that edge does not affect the adjacency of the people sitting at the table.

 

[songoku]

Look up the definition of a cyclic permutation. What does your first statement about a circular table mean?

Arranging objects in circle? Or maybe more appropriate definition is arranging objects in closed shaped so there are several similar arrangements / mirror images?

Why should the shape of the table make a difference? The table has a continuous edge. Putting corners in that edge does not affect the adjacency of the people sitting at the table.

Ok, then does the number of seats matter? Let say 6 people sit on the longer sides of the table, 3 people on each side and no seats on the shorter side so only 6 people and 6 chairs. Is this cyclic permutation or maybe not because they do not form "closed" shape since no chair on shorter side?

Thanks

 

[songoku]

Ok, then does the number of seats matter? Let say 6 people sit on the longer sides of the table, 3 people on each side and no seats on the shorter side so only 6 people and 6 chairs. Is this cyclic permutation or maybe not because they do not form "closed" shape since no chair on shorter side?

Thanks

Additional question that I am also confused about. If the above case is not cyclic permutation, what about if we add, let say 2 chairs on each of the shorter side? Will this become cyclic permutation?

Thanks

 

[Stephen Tashi]

Suppose 6 people want to sit in circular table. This case is considered as cyclic permutation.

No, technically it isn't.

The term "cyclic permutation" has a technical definition. You wont' encounter this technical definition unless you are studying "permutations" in an advanced course.

In elementary courses, as "pemutation" is an "ordered arrangement" - i.e. it is a static thing. In advnced courses a "permutation" is a function. A cyclic permutation is a particular type of function. If you are taking an advanced course we can discuss "cyclic permutations". However, I think what you want to ask about is "circular arrangements". not cyclic permutations.

How about if the 6 people want to sit around a rectangular table?

If you are looking for a rule that tell you how to work word problems in combinatorics, this is not primarily a mathematical question. Authors who invent word problems about people sitting at a table may or may not intend that the shape of the table affects the answer.

The most commonly used terminology in combinatorial word problems is "number of ways". However, there is no standard definition for what a "way" is. In each problem you must decide what the author means by a "way". - i.e. what properties make two "ways" the same way or different ways..

You questions concern whether various arrangements are counted as different "ways". There is no general answer to this. It depends on interpreting the intentions of the author who writes a particular problem.

 

[songoku]

No, technically it isn't.

The term "cyclic permutation" has a technical definition. You wont' encounter this technical definition unless you are studying "permutations" in an advanced course.

In elementary courses, as "pemutation" is an "ordered arrangement" - i.e. it is a static thing. In advnced courses a "permutation" is a function. A cyclic permutation is a particular type of function. If you are taking an advanced course we can discuss "cyclic permutations". However, I think what you want to ask about is "circular arrangements". not cyclic permutations.

No I do not take advanced class What I mean is just exactly what you wrote: circular arrangements

If you are looking for a rule that tell you how to work word problems in combinatorics, this is not primarily a mathematical question. Authors who invent word problems about people sitting at a table may or may not intend that the shape of the table affects the answer.

The most commonly used terminology in combinatorial word problems is "number of ways". However, there is no standard definition for what a "way" is. In each problem you must decide what the author means by a "way". - i.e. what properties make two "ways" the same way or different ways..

You questions concern whether various arrangements are counted as different "ways". There is no general answer to this. It depends on interpreting the intentions of the author who writes a particular problem.

Yeah, this is exactly what I am looking for.

I had test and such question came up and me and my friend interpreted the question differently. Let give some numbers so we can discuss further.

1. Suppose a rectangular table only has 6 seats, 3 on each of the longer sides (no chair on the shorter side). Six people want to sit down. Find how many ways it can be done?
My answer = 6!

2. Suppose two chairs are added, one chair on each of the shorter side. Find number of ways 6 persons can sit down
My answer = 8P6

But my friend gave me this link: https://www.mbatious.com/topic/331/seating-arrangement-around-various-geometrical-figures
In the link, circular arrangement is used to solve the problem.

3. Suppose one particular person want to sit on the shorter side, find number of ways this can be done.
Is this considered as circular arrangement?

Please share how you would tackle the problems. Thanks

 

[Stephen Tashi]

Please share how you would tackle the problems.

Normally, I wouldn't tackle them! Solving word problems in combinatorics tends to be an exercise in literary interpretation - similar to working crossword puzzles or answering riddles. My practical mind thinks it's a waste of time. However, there appear to be some countries where puzzles become standard fare on important examinations. Fortunately, I don't have to take such exams. If you want to be an expert on combinatorial problems involving people-sitting-at-tables-of-particular-shapes, I suppose you have to learn the traditional ways of interpreting that type of problem.

1. Suppose a rectangular table only has 6 seats, 3 on each of the longer sides (no chair on the shorter side). Six people want to sit down. Find how many ways it can be done?
My answer = 6!

That's correct if we consider the seating positions distinguishable. By your definition of the problem, for persons A,B,C,D,E,F, the following arrangements are different "ways"

way 1:
A B C
|table|
D E F

way 2:
D E F
|table|
A B C

There are tarditional problems about circular tables, people sitting in a circle around a campfire etc. With a circular table way 3 and way 1 would be "the same" way.

way 3
B C F
(table)
A D E

Since the problem mentions a rectangular table, we suspect the author wishes to consider way 1 and and way 3 to be different ways.

Looking at the solution claimed in the link you posted, the author considers way 4 to be different than way 1.

way 4
B C A
|table|
F E D

For example, if person A is seated first, the author thinks it's important whether A sits on the left or right of a row of 3 chairs. However, he thinks that way 1 and way 2 are "the same way", so he doesn't think it matters which row of 3 chairs A sits in.

2. Suppose two chairs are added, one chair on each of the shorter side. Find number of ways 6 persons can sit down
My answer = 8P6

Is "8P6" your notation for $\binom{8}{6} = \frac{8!}{6! 2!}$ ?

 

[songoku]

I am really sorry for late reply.

Normally, I wouldn't tackle them! Solving word problems in combinatorics tends to be an exercise in literary interpretation - similar to working crossword puzzles or answering riddles. My practical mind thinks it's a waste of time. However, there appear to be some countries where puzzles become standard fare on important examinations. Fortunately, I don't have to take such exams. If you want to be an expert on combinatorial problems involving people-sitting-at-tables-of-particular-shapes, I suppose you have to learn the traditional ways of interpreting that type of problem.

That's correct if we consider the seating positions distinguishable. By your definition of the problem, for persons A,B,C,D,E,F, the following arrangements are different "ways"

way 1:
A B C
|table|
D E F

way 2:
D E F
|table|
A B C

There are tarditional problems about circular tables, people sitting in a circle around a campfire etc. With a circular table way 3 and way 1 would be "the same" way.

way 3
B C F
(table)
A D E

Since the problem mentions a rectangular table, we suspect the author wishes to consider way 1 and and way 3 to be different ways.

Looking at the solution claimed in the link you posted, the author considers way 4 to be different than way 1.

way 4
B C A
|table|
F E D

For example, if person A is seated first, the author thinks it's important whether A sits on the left or right of a row of 3 chairs. However, he thinks that way 1 and way 2 are "the same way", so he doesn't think it matters which row of 3 chairs A sits in.
Is "8P6" your notation for $\binom{8}{6} = \frac{8!}{6! 2!}$ ?

No, by 8P6 I mean permutation. But after reading your reply, I get the general idea. Thank you very much