Find the number of ways of arranging letters of the word MATHEMATICAL.

1.	Find the number of ways of arranging letters of the word MATHEMATICAL. How many of these arrangements have all vowels together?
Answer» 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 = \(\frac{8!\times5!}{2!3!2!}\)

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

Answer»

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 = \(\frac{8!\times5!}{2!3!2!}\)