Question

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

Answer 1

In the word 'MAHARASHTRA' the number of letters is n = 11 of which A repeats 4 times, i.e., p = 4, H repeats twice i.e., q = 2, R repeats twice, i.e., r = 2 and rest are distinct.

∴ the number of different arrangements with the letters of the word MAHARASHTRA is

`("n"!)/("p"!"q"!"r"!)`

= `(11!)/(4!2!2!)`

= `(11 xx 10 xx 9 xx 8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 2 xx 1 xx 2 xx 1)`

= 11 × 10 × 9 × 2 × 7 × 6 × 5

= 415800

Vowels are A, A, A, A always together.

The number of arrangements of 4 vowels = `(4!)/(4!)` = 1  ...(∵ A repeats 4 times.)

Considering 4 vowels as one unit (object), we have altogether 7 + 1 = 8 letters, i.e., n = 8 of which H repeats twice, i.e., p = 2 and R repeats twice, i.e., q = 2.

∴ number of possible arrangements of the letters of the word MAHARASHTRA in which all the vowels are together.

= `(8!)/(2!2!) xx 1`

= `(8!)/(2!2!)`

= `(8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1)/(2 xx 1 xx  2 xx 1)`

= 10080