- 10080
- 120960
- 4989600
- 21160
- None of these
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Answer: Option 2
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
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