In how many different ways can the letters of the word mathematics be arranged?

2 i believe would be the answer

Treat the vowels like they are 1 block, as if they are one letter. But in this vowel block there are 4 vowels, 2 of which are the same. The number of ways of arranging these 4 vowels is then 4! 2! 4! 2!.

Now, treating this block aa one letter, we have 7 consonants and 1 block, so there are 8 'letters' to arrange with two M's and two T's. So we have to divide out by 2! for each. So we get: 8! 2!2! 8! 2!2!.

Finally we multiply this by the number of ways the vowels can be arranged to get:

4! 2! 8! 2!2!