Question

Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have

(a) letters R and H never together?

(b) all vowels together?

Answer 1

There are 11 letters in the word MAHARASHTRA in which ‘A’ is repeated 4 times, ‘H’ repeated 2 times, and ‘R’ repeated 2 times.

∴ Total number of words can be formed = \(\frac{11!}{4!2!2!}\)

(a) When letters R and H are never together. Other than 2R, 2H there are 4A, 1S, 1T, 1M. These letters can be arranged in \(\frac{7!}{4!}\) ways = 210

These seven letters create 8 gaps in which 2R, 2H are to be arranged.

Number of ways to do = \(\frac{^8P_4}{2!2!}=420\) 

Required number of arrangements = 210 × 420 = 88200.

(b) When all vowels are together. There are 4 vowels in the word MAHARASHTRA, i.e., A, A, A, A. Let us consider these 4 vowels as one unit, which can be arranged among themselves in \(\frac{4!}{4!}\) = 1 way.

This unit is to be arranged with 7 other letters in which ‘H’ is repeated 2 times, ‘R’ is repeated 2 times.

 ∴ Total number of arrangements = \(\frac{8!}{2!2!}\)