Question

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

Answer 1

total letters = 11
vowels = 4 (2 are same)
no. of ways they can be arranged = 4!/2! = 12

no of consonants = 7(2 M's, 2 T's,)
Assume all the vowels as 1 quanity

so total = 7 consonants + 1 vowel quantity = 8

total ways = $\frac{8!*4!}{(2!*2!)*(2!)} =120960$

So total 120960 ways.