12 letters choose 7, that C(12,7) = 792 ways to choose 7 letters
for somebody to sit at. 

Notice that order DOES NOT matter when picking the letters,
because picking letters J,A,C,G,I,E,K is the same as picking
the letters C,K,I,J,A,G,E.


 For each of those 792 ways to pick 7
letters, the 7 people can be arranged in P(7,7) or 7! = 5040 ways.

Notice that here order DOES matter when choosing which person sits
where, because if two people switch seats, that is a different
seating arrangement.

Answer = C(12,7)×P(7,7) = 792×5040 = 3991680

[In case that's not the right answer, it's because it was assumed that
it was a round table mounted on a turn-table floor, so that no matter how
you turned the people around, it would not count as a different seating
arrangement, so the answer would be the above answer divided by the 12
ways the rotating floor could be turned.]

Edwin