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

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

A. 10080

B. 4989600

C. 120960

D. None of these

### **Answer : ** C. 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!)(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**.