Question

Harry, Peter and 7 girls sit in a row. Harry can't sit in an end seat. In how many ways can they be arranged?

Answer 1

All possibilities = 9! 
possibilities that harry sits at an end = 8! + 8! 
possibilities that harry doesn't sit at an end = 9! - 8! -8! 

Answer 2

List of all possibilities, without restrictions (ie Harry can sit in any of the 9 seats): $9!$
List of all possibilities where Harry sits on either end: $2 \cdot 8!$

Hence, list of all possibilities where Harry doesn't sit on either end: $9! - 2 \cdot 8! = 282,240$ ways.