Permutation

[Keith5262]

There are 8 chairs around a table. One chair is reserved for the President. How many seating arrangements are possible for the President and his 7 advisors?

 

[Deleted member 4993]

There are 8 chairs around a table. One chair is reserved for the President. How many seating arrangements are possible for the President and his 7 advisors?

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[BigBeachBanana]

There are 8 chairs around a table. One chair is reserved for the President. How many seating arrangements are possible for the President and his 7 advisors?

If the President's chair is fixed in one spot, then you'll have to consider arranging [imath]n=8[/imath] objects in a circular formation.
See Circular Permutation.

 

[Deleted member 4993]

If the President's chair is fixed in one spot, then you'll have to consider arranging [imath]n=7[/imath] objects in a circular formation.
See Circular Permutation.

After we fix a chair (at a spot) for a particular person , I do not see this situation as circular permutation.

 

[BigBeachBanana]

After we fix a chair (at a spot) for a particular person , I do not see this situation as circular permutation.

Why do you say that? Curious as to how you see the problem.
Considering the case with four people instead (n=4). We fix the President's chair at (1). It follows that the number of arrangements is [imath](4-1)!=3!=6[/imath] as demonstrated by Wolfram Alpha.

 

[Harry_the_cat]

After we fix a chair (at a spot) for a particular person , I do not see this situation as circular permutation.

I agree with Subhotash.

And your solution above is really just the number of permutations of 3 things. The "circular" arrangement (once the 1 is placed) is irrelevant. You may as well just list the 6 permutations as 432, 342, 423, 243, 324, 234.

 

[pka]

Why do you say that? Curious as to how you see the problem.
Considering the case with four people instead (n=4). We fix the President's chair at (1). It follows that the number of arrangements is [imath](4-1)!=3!=6[/imath]

Actually this is not a circular arrangement. If we seat eight people at a circular table can be done in [imath](8-1)!=7![/imath] ways. That is true of any circular permutation. However, if there is an assigned seat, as in this case, that makes it an ordered table as opposed to an unordered table.
That means there are seven other people left to be seated which can be done in [imath]7![/imath] ways. Lets explore a different situation.
Lets say we have the eight people among whom is a president and a vice-president. These two are seated opposite to each other in assigned seats.
That means there are six people left to be seated which can be done in [imath]6![/imath] ways. A circular permutation does not apply in either case.

 

[BigBeachBanana]

After reviewing your comments, and my notes on circular permutation, I concluded that arranging n objects around a table would not fit the definition of a circular permutation, even when one of the seats wasn't assigned.

By the definition of circular permutation, "the arrangement of things in a circle or a ring is called circular permutations. The fundamental difference between linear and circular permutations is that in the former, there are always two separate ends, but in circular permutations, we cannot distinguish the two ends. For this, in linear permutations, arrangements depend on the absolute position while in the case of circular permutations, we shall be concerned with relative positions of the things. Thus, no. of circular permutations of n different things taken all at a time is (n−1)! ways taking one of the n things fixed."

Ignoring assigning a chair, there is a very subtle but paramount difference between "How can you arrange 8 people in a ring?" versus "How can you arrange 8 people around a table?" Two ways are identical for the “ring" question if each individual has the same left and right neighbors. In contrast, in the "table" question, two "ways" are identical if each individual has the same left and right neighbors AND the same place at the table.

With that being said, the "table" question has n possible rotations; thus, the answer to the "table" question will be n(n-1)!, where (n-1)! is the answer to the "ring" question. (Ignoring the assigned seat)

 

[Harry_the_cat]

BBB, I agree. I've always questioned the wording and answers to those typical textbook questions on circular arrangements around a table for the exact reason you mention.

 

[pka]

I agree with Subhotash. And your solution above is really just the number of permutations of 3 things. The "circular" arrangement (once the 1 is placed) is irrelevant. You may as well just list the 6 permutations as 432, 342, 423, 243, 324, 234.

BBB, I agree. I've always questioned the wording and answers to those typical textbook questions on circular arrangements around a table for the exact reason you mention.

To Harry, I cannot see how those two posts refer to the same issue. Honestly they seem completely contradictory to me.
I have chewed on the most of the day. I just do not want such gross miss-information on this forum.
Here is a good reference by the great Ivan Niven who was student of LEDickon at Chicago; his pedigree could not be better.
Niven's small text MATHEMATICS OF CHOICE, how to count without counting, in the MAA"s New Mathematical Library is now a classic.
If I read both BBB & Harry'S comments correctly they both would apply to Ivan's discussion on page 22
Section 2.6 Permutations of Things in a Circle. Now any talk of a difference in a round table over-against a ring is that of an utter armature.
Any mathematics library worth its name will have this book. Here are two questions pertinent to this thread: page 24, set 6, #2 & 3.
2) If eight people are to be seated around a table among which two who are not allowed to be seated next to one another, how many ways are there to seat them? ANSWER 3600 How?

3) Those eight people consist of four men & four ladies. How many ways are there to seat them at a round table if no two men are to be in adjacent seats? ANSWER: 144 How?

 

[Harry_the_cat]

Mmmm. With all due respect, I'm not sure why you find those two statements contradictory. But let me address them separately.

Statement 1: "... your solution above (see Post #5) is really just the number of permutations of 3 things. The "circular" arrangement (once the 1 is placed) is irrelevant. You may as well just list the 6 permutations as 432, 342, 423, 243, 324, 234."

I think you agree with this, because your solution to the original post (where there are 8 people with one already seated) is 7! applying the same logic.

Your post #7 says:

Actually this is not a circular arrangement. If we seat eight people at a circular table can be done in (8-1)!=7! ways. That is true of any circular permutation. However, if there is an assigned seat, as in this case, that makes it an ordered table as opposed to an unordered table. That means there are seven other people left to be seated which can be done in 7! ways.

So here you are saying that the number of ways of sitting 8 people at a table (with no restrictions) is the same as the number of ways of sitting 8 people at a table where one person has an assigned seat (ie sitting 7 people at a table when one person is already in an assigned seat). That seems contradictory to me.

I do agree with the answer (ie 7!), which is just the number of permutations of 7 things.

Statement 2: "BBB, I agree . I've always questioned the wording and answers to those typical textbook questions on circular arrangements around a table for the exact reason you mention (see post #8)."

I suppose this comes down to the definition of "permutation" and "arrangement" which are often used interchangeably (incorrectly in my opinion) in textbook questions.

Your post #10 says:

Now any talk of a difference in a round table over-against a ring is that of an utter armature.

When a table is mentioned, it is in a context, eg in a room. It is not merely theoretical any longer. Consider the following scenario of a table in a room with a window.

I could concede that the "permutation" (order) of seats around the table is the same - in both cases for example, B is to the left of A, etc.
BUT, I would argue that the "arrangement" of seats around the table is not the same. In the first scenario, A has his/her back to the window while in the second scenario A has a view out the window. If you were person A and didn't want the bright light coming in at your face (in scenario 2), then you would prefer scenario 1. Because one is preferable to the other, they must be different.

If, however, they were literally rings with an Amethyst, a Birthstone, a Cubic zirconia and a Diamond, then they would be the same, simply because of the rotatability of the ring.

I don't think what I've said is of "an utter armature" (or even amateur). I also don't think it is "such gross miss-information" but is a good topic for discussion.