In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

Answer: Option C

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
∴ Number of ways of arranging these letters = $$\frac{8!}{\left(2!\right)\left(2!\right)}$$   = 10080
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = $$\frac{4!}{2!}$$ = 12
∴ Required number of words = (10080 x 12) = 120960