Question

Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have all vowels together?

Answer 1

Total number of letters in the word MAHARASHTRA = 11
The letter ‘A’ is repeated ‘4’ times.
The letter ‘H’ is repeated twice.
The letter ‘R’ is repeated twice.

∴ Number of arrangements = `(11!)/(4!2!2!)`

Here, all vowels are together. The given word has 4 vowels A, A, A, A.
We consider 4 vowels together as a single letter, say G.
We have 8 letters G, M, H, R, S, H, T, R of which R and H are repeated 2 times each.
The number of ways of arranging 8 letters

= `(8!)/(2!2!)`
After this is done, 4 vowels (in which A is repeated 4 times) can be arranged in `(4!)/(4!)`= 1 way.
∴ The number of arrangements in which the vowels are together = `(8!)/(2!2!)`