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

Answer: Option B

Keeping the vowels (AEIA) together, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which we have 2M, 2T and the rest are all different.
Number of ways of arranging these letters
$$\begin{array}{rl}& =\frac{8!}{2!.2!}\\ & =\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1\times 2\times 1}\\ & =10080\end{array}$$
Now, (AEAI) has 4 letters, out of which we have 2A, 1E and 1I.
Number of ways of arranging these letters
$$\begin{array}{rl}& =\frac{4!}{2!}\\ & =\frac{4\times 3\times 2\times 1}{2}\\ & =12\end{array}$$
∴ Required number of ways = (10080 × 12) = 120960