I'm not sure if I made illegal maths moves in the way I did it but I'll explain my thought process:
Obviously approximate via normal distribution, and place the mean as 17.5 (expected value of extra seats the airline can sell). The standard deviation was something like 4.08, and to ensure only 1% of flights can be over booked the z score was like 2.33 (for a 0.99 probability of success).
Just put those values into the z score formula then solve for x, which comes out to be 8; thus the airline can sell 358 seats.
It seemed a lot simpler than the answers hence my confusion but it seems okay to me? That's how I did it