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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?

1. $10080$
2. $4989600$
3. $120960$
4. None of the options

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## 2 Answers

+1 vote

Option 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 =    $\frac{8!}{(2!)(2!)}$    = 10080.

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

Number of ways of arranging these letters =    $\frac{4!}{(2!)}$    = 12.
Required number of words = (10080 x 12) = 120960.

by (690 points) 1 3 17
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$\underbrace{MTHMTCS\underbrace{AEAI}_{\frac{4!}{2!} = \frac{24}{2} = 12}}_{\dfrac{8!}{2!2!} \times 12} = 120960.$

So, the correct answer is $(C).$
by (5.8k points) 5 56 296
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