# Permutations and combinations

1. Nov 28, 2013

### MSG100

First of all, I would like to apologize for my bad english.

The Problem:

There is n members inclusive CEO and 2 vice presidents. In how many ways can they be seated around a table so that both vice presidents sits next to the CEO?

Attempt at a solution:

There's in total (n+2)! ways to be seated at the table.

It should be divided by a vice president to the left (if his place are fixed). That gives us (n+1) opportunities. The same on the left hand side.

The other vice president only have one side to choose when the CEO and vice president are sat, (n).

All in all:

(2(n+2)!)/((n+1)(n))

Is this the right way to think?

2. Nov 28, 2013

### haruspex

It says 'inclusive', so there are only n altogether. You should not be getting an n+2 term.
I'm afraid I was not able to follow your reasoning from there.
You could go through these steps: how many ways to place the CEO? Now how many to place the VPs? Now how many to place everyone else?

3. Nov 28, 2013

### MSG100

Ok, I don't think your alone not undertand what I just write:)
I need to do some more basic stuff about permutations and then come back to this problem.

4. Nov 29, 2013

### HallsofIvy

Staff Emeritus
Your English is excellent except for the specific question of whether there are n people, with the CEO and vice presidents included, or whether there are n people plus the CEO and vice-presidents, making a total of n+ 2 people.

However many "other" people there are, I would start by seating the CEO. Since this is a round table with all seats equivalent, any seat will do. We then seat the two vice presidents beside the CEO. There are 2 ways to do this: calling one vice presidednt "A" and the other "B", "A on the CEO'S left and B on the right" or "A on the CEO's right and B on the left". Once we have seated them, we seat the remaining "m" (whether m= n or m= n--3) people- there are m! ways to do that.

5. Nov 29, 2013

### Ray Vickson

There is a remaining issue, which is whether or not we regard all starting places of the CEO as equivalent, or whether we should multiply your suggested answer by n. One could argue either way; it depends on whether, for example, you regard the arrangements in which the CEO is in the northwest seat as being different from that where he/she is in the southeast seat, etc.

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