# 12 persons are to be arranged to a round table if two particular person among them are not to be side by side the total number of arrangement is?

Jul 3, 2018

90*9! =32,659,200 arrangement.

#### Explanation:

As far as I can see, we can find it this way:
Let's start with one of the two particular persons. On one side we have 10 persons we can choose among. On the other we can choose among 9 persons, 90 persons in all.
When these three persons are chosen, we have 9 left to distribute, which can be done in 9! different ways.

So my solution would be 90*9! =32,659,200 different arrangements.

9xx10! = 9xx"3,628,800"="32,659,200"

#### Explanation:

I'll use an alternate way of tackling this question and see if I end up with @Roy O's answer.

First we need to remember that this is a round table and so there is no starting seat or ending seat - we're only looking at seating relationships between people.

So let's first seat persons A and B - they are not to sit next to each other.

We can put A into any seat - but since there isn't a starting seat or ending seat, we'll just put A into a seat that I'll simply call A's seat. (If we were sitting in a row, we'd start by saying that there are 12 seats A can be in, but here in a circle we'll skip this).

Now for B. B can't sit next to A. A is taking up one seat and there are the two seats (one to either side) that are next to A, so 3 seats in total where B can't sit. How many seats are then available for B? 9.

And now the remaining 10 people can sit anywhere they desire in the remaining seats. They can sit in 10! ways.

This all gives:

9xx10! = 9xx"3,628,800"="32,659,200"