Question

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 1

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.