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Find how many arrangements can be made with the letters of the word “MATHEMATICS” in which the vowels occur together?

(a) 10080 

(b) 120960 

(c) 14400 

(d) 36500

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(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.

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