Discussion

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

Answer: C

MATHEMATICS has eleven letters, with M, A and T each appearing twice. Its vowels are A, E, A, I and its consonants M, T, H, M, T, C, S.

Treating the four vowels as one block gives eight units, among which M and T each repeat:

\[ \frac{8!}{2!\,2!} = \frac{40320}{4} = 10{,}080 \]

Within the block the four vowels include two A's:

\[ \frac{4!}{2!} = 12 \]

\[ 10{,}080 \times 12 = 120{,}960 \]

Three separate repetitions have to be divided out - the M's and T's among the units, and the A's inside the block.

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