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

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together? Correct Answer 120960

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
∴ Number of ways of arranging these letters = $$\frac{{8!}}{{\left( {2!} \right)\left( {2!} \right)}}$$   = 10080
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = $$\frac{{4!}}{{2!}}$$ = 12
∴ Required number of words = (10080 x 12) = 120960

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