Question

Find the number of ways of arranging letters of the word MATHEMATICAL. How many of these arrangements have all vowels together?

Answer 1

There are 12 letters in the word MATHEMATICAL in which ‘M’ is repeated 2 times, ‘A’ repeated 3 times and ‘T’ repeated 2 times.

∴ Required number of arrangements = \(\frac{12!}{2!3!2!}\) 

When all the vowels,

i.e., ‘A’, ‘A’, ‘A’, ‘E’, ‘I’ are to be kept together.

Let us consider them as one unit.

Number of arrangements of these vowels among themselves = \(\frac {5!}{3!}\) ways.

This unit is to be arranged with 7 other letters in which ‘M’ and ‘T’ repeated 2 times each.

∴ Number of such arrangements = \(\frac{8!}{2!2!}\) 

∴ Required number of arrangements = \(\frac{8!\times5!}{2!3!2!}\)