(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`

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