Find the number of ways to arrange the letters of the word 'MATHEMATICS' such that the vowels are always together.

Solution

1. $Thewor{d}^{\prime }MATHEMATIC{S}^{\prime }contains11letters:M,M,A,T,T,H,E,A,I,C,S,withvowelsA,A,E,IandconsonantsM,M,T,T,H,C,S.$
2. Treat the four vowels as a single block, leaving 7 consonants and 1 vowel block, totaling 8 units to arrange, where M and T repeat twice: $\frac{8!}{2!2!}=10080$.
3. Arrange the vowels {A, A, E, I} internally, where A repeats twice: $\frac{4!}{2!}=12$.
4. Multiply the arrangements of the units by the internal arrangements of the vowels: $10080\times 12=120960$.
5. Hence the answer is (A).