Eg 1. There are 10 seats in a circle and three are empty. How many ways can this be arranged? $(10-1) !=9 !$
Answer
Once the three empty seats have been chosen, the remaining people can be placed in $7!$ ways. Therefore, the total number of ways for the 10 seats to be arranged is $2\cdot4\cdot7!+4\cdot3\cdot7!=\boxed{30240}$.
Step 1 ：First, we can choose a seat for the first empty seat. It doesn't matter which seat we choose because we can rotate the table to move the first empty seat to wherever we want. After the first empty seat has been chosen, there are 6 seats left for the second empty seat. Of these seats, 2 are two seats away from the first empty seat, and 4 are not.
Step 2 ：If the second empty seat is two seats away from the first empty seat, there will be 4 places left for the third empty seat. If the second empty seat is in one of the other seats, there will be 3 places left for the third empty seat.
Step 3 ：Once the three empty seats have been chosen, the remaining people can be placed in $7!$ ways. Therefore, the total number of ways for the 10 seats to be arranged is $2\cdot4\cdot7!+4\cdot3\cdot7!=\boxed{30240}$.
