Q. 283.8( 6 Votes )

The letters of the word ‘ZENITH’ are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word ‘ZENITH’?

Answer :

Given the word ZENITH. It has 6 letters.

To find: Total number of words that can be generated by relative arranging the letters of the word ZENITH.


Since it has 6 letters with no repetition, therefore the number of ways of arranging 6 letters on 6 positions is 6! = 720


To find: Rank of word ZENITH when all its permutations are arranged in alphabetical order, i.e. in a dictionary.


First comes, the words starting from the letter E = 5! = 120


words starting from the letter H = 5! = 120


words starting from the letter I = 5! = 120


words starting from the letter N = 5! = 120


words starting from the letter T = 5! = 120


words starting from letter Z:


words starting from ZE:


words starting from ZEH = 3! = 6


words starting from ZEI = 3! = 6


words starting from ZEN:


words starting from ZENH = 2! =2


words starting from ZENI:


words starting from ZENIHT = 1


ZENITH = 1


The rank of word ZENITH = 120 + 120 + 120 + 120 + 120 + 6 + 6 + 2 + 1 + 1


= 616


Hence, the rank of the word ZENITH when arranged in the dictionary is 616.


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