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

10080

4989600

120960

None of these

Explanation:

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.

$\therefore$ Number of ways of arranging these letters =$ \dfrac{8!}{(2!)(2!)} $= 10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters =$ \dfrac{4!}{2!} $= 12.

$\therefore$ Required number of words =$\left (10080 \times 12\right)$ = 120960.