A (small) theatre contains 10 seats. The seats are numbered from 1 to 10, but not in order. Alice, Bob and Carol have tickets for seats 1, 2 and 3 respectively, and Damien, Eve, Francis, and Gertrude each have tickets without an assigned seat. Unfortunately they all arrive late and the lights are dim, so they can’t see what seat they are sitting in.
a) In how many ways can they be seated?
b) In how many ways can they be seated such that Alice is in her correct seat (eg, seat number 1)?
c) In how many ways can they be seated such that Alice, Bob and Carol are all in their correct seats?
d) In how many ways can they be seated such that none of Alice, Bob, Carol are in their correct seats?
What I did for part A is The first person has 10 seats to choose from, then the second person has 9 and so on. I got $10\times9\times8\times7\times6\times5\times4$ as the answer.
For part B I said alice has 1 seat to choose from and then did the rest like part A so I got $1\times9\times8\times7\times6\times5\times4$.
For C I did the same as B but with 3 people and got $1\times1\times1\times7\times6\times5\times4$ but this doesn't seem right to me.
A bit confused on how to do the rest.