The number of ways to arrange the letters of the word 'MATHEMATICS' such that the vowels always come together is:

Solution

1. $Thewor{d}^{\prime }MATHEMATIC{S}^{\prime }contains11letters:vowelsA,A,E,IandconsonantsM,M,T,T,H,C,S.$
2. Treating the four vowels as a single block, we have 8 items to arrange (the block plus 7 consonants), which can be done in $\frac{8!}{2!2!}$ ways because M and T repeat twice each.
3. The four vowels within the block can be arranged in $\frac{4!}{2!}$ ways because A repeats twice.
4. The total number of arrangements is $$\begin{array}{rl}\frac{8!}{2!2!}\times \frac{4!}{2!}& =10080\times 12\\ & =120960\end{array}$$.
5. Hence the answer is (A).