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’?

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.