Question

In how many different ways can the

Answer 1

My answer isn't in your choices so it's probably wrong, but hopefully my approach can give you hints.

The vowels are AEAI
The vowels can be arranged in 4*3*2=24 ways.

The non vowels are MTHMTCS
If the vowels must be together, there have to be in one of the spaces between the non vowels.
If we look at it like this:
-M-T-H-M-T-C-S-
The 4 vowels must go in one of the places marked with a dash.
There are thus 8 places to choose from.

But we must also consider that the non vowels can be arranged in any way. There are 7 of them, so they can be arranged in 7*6*5*4*3*2=5040 ways.

Multiply everything together to get 8*5040*24=967680