Question

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?

Answer 1

`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.

Answer 2

`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"`