In how many different ways can the letters of the word 'MATHEMATICS' be arranged

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Question
In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?
Options
A. 120960
B. 120000
C. 146700
D. None of these

Correct Answer

120960

Explanation

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 = 8!/(2! x 2!)= 10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters =4!/2!= 12.

Required number of words = (10080 x 12) = 120960