In how many ways can 8 people be seated in a row of chairs if two of the people, John and Paul, refuse to sit next to each other?
Well I'm really tired but here we go, let us use complementary counting, so that is
$total$ $outcomes$ - unfavorable$ $outcomes$, total outcomes is all ways that can be seated without restrictions so that is
8!= $40320$, then the unfavorable outcomes is if John and Paul sit next to each other, we can pretend that the pair is one person and that is how many ways to arrange 7 people, which is 7!, that results in $5040$
Subtracting, we get:
$40320-5040$ = $35280$
Please check, as I possibly have missed somethings but that's the idea.
I have no idea if the above answer is correct.....hope it is
here is my take on it
there are 8! seating arrangements = 40320
there are 14 ways those two knuckleheads could be seated next to each other ( you can reverse their order to get 14)
those 14 ways leave 6! for the rest of the folks to be seated
40320 - 14 * 6! = 30240 ways
I do not know if this is correct.....it MIGHT be...it might NOT be !
Let's suppose that they can sit together
John and Paul can occupy any of 7 positions and for each of these,they can sit in 2 ways...and for all these arrangements the other people can sit in 6! ways
So....the total arrangements where they sit together = 7 * 2 * 6! = 10080 ways
And the total possible arrangements = 8! = 40320
So....the ways which they don't sit together = total arrangements - those where they sit together = 40320 - 10080 = 30240