(a) This is just 8 people being arranged in a row: 8! = 40,320

(b) Regard the 2 boys as one "unit" and so there are 7 "units" to arrange. This can be done 7! = 5040 ways.

The boys can be arranged in 2! = 2 ways, so the required answer is

7! × 2! = 10,080

(c) There are only 2 possibilities: the boys are together or they are not.

So the number of ways of arranging so that the boys are not together is:

40,320 − 10,080 = 30,240