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Permutation and Combination

Q1: In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

A 10080

B 4989600

C 120960

D None of these

ANS:C - 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 = 8! = 10080.
(2!)(2!)
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = 4! = 12.
2!
 Required number of words = (10080 x 12) = 120960.



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