-- One zone, consisting of exactly two people, the teacher and the difficult student. Their identities don't change, and their arrangement doesn't change.
-- The other zone, consisting of the other 9 students. They can line up in any possible way.
How many ways can you line up 9 students ?
The first one can be any one of 9. For each of these . . . The second one can be any one of the remaining 8. For each of these . . . The third one can be any one of the remaining 7. For each of these . . . The fourth one can be any one of the remaining 6. For each of these . . . The fifth one can be any one of the remaining 5. For each of these . . . The sixth one can be any one of the remaining 4. For each of these . . . The seventh one can be any one of the remaining 3. For each of these . . . The eighth one can be either of the remaining 2. For each of these . . . The ninth one must be the only one remaining student.
The total number of possible line-ups is
(9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) = 9! = 362,880 .
But wait ! We're not done yet !
For each possible line-up, the teacher and the difficult student can sit
-- On the left end, -- Between the 1st and 2nd students in the lineup, -- Between the 2nd and 3rd students in the lineup, -- Between the 3rd and 4th students in the lineup, -- Between the 4th and 5th students in the lineup, -- Between the 5th and 6th students in the lineup, -- Between the 6th and 7th students in the lineup, -- Between the 7th and 8th students in the lineup, -- Between the 8th and 9th students in the lineup, -- On the right end.
That's 10 different places to put the teacher and the difficult student, in EACH possible line-up of the other 9 .
So the total total number of ways to do this is
(362,880) x (10) = 3,628,800 ways.
If they sit a different way at every game, the class can see a bunch of games without duplicating their seating arrangement !
