Q. 24

# If the letters of the word ‘MOTHER’ are written in all possible orders and these words are written out as in a dictionary, find the rank of the word ‘MOTHER’.

Given the word MOTHER.

Arranging the permutations of the letters of the word MOTHER in a dictionary:

To find: Rank of word MOTHER in dictionary.

First comes, words starting from letter E = 5! (5 letters no repetation) = 120

words starting from letter H = 5! (5 letters no repetation) = 120

words starting from ME = 4! (4 letters no repetation) = 24

words starting from MH = 4! (4 letters no repetation) = 24

words starting from MOE = 3! (3 letters no repetation) = 6

words starting from MOH = 3! (3 letters no repetation) = 6

words starting from MOR = 3! (3 letters no repetation) = 6

words starting from MOTE = 2! (2 letters no repetation) = 2

words starting from MOTH:

MOTHER = 1

Rank of the word MOTHER = 120 + 120 + 24 + 24 + 6 + 6 + 6 + 2 + 1

= 309

Hence the rank of the word MOTHER in arranging the letters of MOTHER in a dictionary among its permutations is 309.

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