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