Counting

There are 18  padded chairs around a circular table, and the chairs are numbered from 1  through 18 . How many ways can five people take their seats, so that no two people are adjacent?

let each person have a chair, no number yet, and also take a empty chair to the right, tie the chairs together

now there are 5 'fat' chairs and 8 chairs not used yet.

So that is  13 altogether. 

 the circle part is a little confusing but rotations are not the same as they usually ars so in most respectes this can be thought of as a line

I was thinking  13C5 to choice the people seats times by 5! to order them.     13C5 * 5!

but in this senario chair 18 will never be sat in.

ok now I will put someone specifically in chair 18  and then I will delete that chair and that person from the scenario.

So 17 chairs left and 4 people left

So now I have 4 'fat' people and and 9 cother chair        1 + 4*2 + 9 =18

the people  be arranged in   13C4 * 4! ways

So i think the answer is     13C5*5!  +  13C4 *4! = 154440+ 17160 = 171 600 ways