The letters of the words 'ZENTH' 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 'ZENTH'?

In a dictionary the words at each stage are arranged in alphabetical order. In the given problem we must therefore consider the words beginning with E, H, I, N, T, Z in order.

'E' will occur in the first place as often as there are ways of arranging the remaining 5 letters all at a time i.e. E will occur 5! times. Similarly H will occur in the first place the same number of times.

$\therefore$ Number of words starting with E= 5!

=$5\times 4\times 3\times 2\times$ 1= 120

Number of words starting with H=5! = 120

Number of words skirting with I=5! = 120

Number of words starting with N=5! = 120

Number of words starting with T=5! = 120

Number of words beginning with Z is 5!, but one of these words is the word ZENITH itself.

So, we first find the number of words beginning with ZEH, ZEI and ZENH

Number of words starting with ZEH = 3!
=6

Number of words starting with ZENH = 2!
=2

Now, the word beginning with ZENI must follow.

There are 2! words beginning with ZENI is the word ZENIHT and hte next word is ZENITH

$\therefore$ Rank of ZENITH

= $5\times 120+2\times 6+2+2$

=600+12+4

= 600+16 = 616