The following problem:
Ten diplomatic delegates are seated in a row. There are two specific seating requirements: 1) France and Britain are sat next to each other, and 2) the U.S. and Russia are sat at least one seat away from one another. How many seating arrangements are there?
Is in the section for multinomial coefficients in my book. I'm not sure how to use the multinomial coefficient to solve this. Actually, I don't know how to solve this, period.
An approach I've tried and failed:
There are actually nine seats, if we treat Britain and France as one seat with two permutations.
Decide whether the U.S. is a side seat (seat numbers 1 or 9) or not.
Consider two cases, one where the U.S. is a side seat, one where it's not. Assign to either one of the two side seats, or to one of the seven middle seats.
Assign Russia to either one of the seven seats remaining, or one of the six seats remaining.
Now two of the nine seats have been taken. Assign Britain and France to one of the 7 remaining seats:
Six seats remain. Assign remaining countries.
The problem with doing the above approach is that the resulting number is off by a magnitude of like, $10^8$. It's obscenely large.
Can I have some help? Where did I screw up?
So, for some reason I added factorials to combinations that did not need any... I'm still off by a whole magnitude, though, so my original problem still stands.
Okay, so actually I get the correct answer if I follow the steps above, I just need to divide by a factor of 2, since I don't need to multiply by 2 to count the end seat scenario... I'm already doing that by addition.
I still have a question, though:
How do I solve this with multinomial coefficients?