Let the word be MATHEMATICS.
The letters are M, A, T, H, E, M, A, T, I, C, S.
The number of letters in the word MATHEMATICS is 11.
The vowels are A, E, A, I.
The consonants are M, T, H, M, T, C, S.
The number of vowels is 4. The vowels are A, E, A, I.
The number of consonants is 7. The consonants are M, T, H, M, T, C, S.
We want to arrange the letters such that the vowels always come together.
Consider the vowels A, E, A, I as a single unit (V).
Then we are arranging the letters M, T, H, M, T, C, S, V.
There are 8 letters in this set.
The number of arrangements of these 8 letters is $\frac{8!}{2!2!}$, since M appears twice and T appears twice.
The number of arrangements of the vowels A, E, A, I is $\frac{4!}{2!}$, since A appears twice.
Therefore, the total number of arrangements of the letters in MATHEMATICS such that the vowels always come together is
$$\frac{8!}{2!2!} \times \frac{4!}{2!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 2} \times \frac{4 \times 3 \times 2 \times 1}{2} = \frac{40320}{4} \times \frac{24}{2} = 10080 \times 12 = 120960$$
Thus, the total number of arrangements of the letters of the word MATHEMATICS such that the vowels always come together is 120960.
The vowels are A, E, A, I.
The consonants are M, T, H, M, T, C, S.
Treat the vowels A, E, A, I as a single entity.
Then we have 8 entities to arrange: M, T, H, M, T, C, S, (AEAI).
The number of arrangements of these 8 entities is $\frac{8!}{2! 2!}$, where the factorials in the denominator are due to the repetitions of the letters M and T.
The number of arrangements of the vowels A, E, A, I is $\frac{4!}{2!}$, where the factorial in the denominator is due to the repeated letter A.
The total number of arrangements is
$$ \frac{8!}{2! 2!} \times \frac{4!}{2!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} \times \frac{4 \times 3 \times 2 \times 1}{2 \times 1} = \frac{40320}{4} \times \frac{24}{2} = 10080 \times 12 = 120960$$
Final Answer: The final answer is $\boxed{120960}$
Prompt: In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?