(b) 120960
There are 11 letters in the word “MATHEMATICS” out of which 4 are vowels and the rest 7 are consonants.
Let the four vowels be written together. A A E I M, T, H, M, T, C, S
Consider the four vowels as one as unit, then these 8 letters (7 consonants and the vowel unit) can be permuted in \(\frac{8!}{2!2!}\) = 10080 ways. (There are two pairs of same letters AA and MM)
Corresponding to each of these permutations, the 4 vowels can be arranged among themselves in \(\frac{4!}{2!}\) = 12 ways.
∴ Required number of word in which vowels occur together = \(\frac{8!}{2!2!}\times\frac{4!}{2!} = 10080\times12\) = 120960.