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

Solution(By Examveda Team)

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
$$\eqalign{ & = \frac{{8!}}{{2!.2!}} \cr & = \frac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \cr & = 10080 \cr} $$
Now, (AEAI) has 4 letters, out of which we have 2A, 1E and 1I.
Number of ways of arranging these letters
$$\eqalign{ & = \frac{{4!}}{{2!}} \cr & = \frac{{4 \times 3 \times 2 \times 1}}{2} \cr & = 12 \cr} $$
∴ Required number of ways = (10080 × 12) = 120960