__Answer with Explanation:__**I**

**n 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 = 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 = 4!/2! = 12.

Therefore, Required number of words = (10080 x x 12) = 120960.

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 = 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 = 4!/2! = 12.

Therefore, Required number of words = (10080 x x 12) = 120960.