Answer: (a) \(282240\); (b) \(120\).

We start with the main method. Count the total arrangements first, then subtract restricted cases or treat grouped letters as a single block.

There are 9 different letters in the word POLYMATHS, so the total number of arrangements is \(9!\).

For part (a), count the arrangements where O and A are together. Treat OA or AO as one block.

There are \(8!\) arrangements of the block and the other 7 letters, and 2 orders inside the block.

So the number with O and A together is \(2\cdot8!\).

Hence the number where O and A are not next to each other is

\(9!-2\cdot8!=362880-80640=282240.\)

For part (b), the letters MATHS must stay together in that exact order, so treat MATHS as one object.

Together with P, O, L and Y, there are 5 objects to arrange.

Therefore the number of arrangements is

\(5!=120.\)

This completes the solution and gives the required result.