How to approach permutation problems like these?

Question

I have a permutation problem here-

"The letters of the word ZENITH are written in all possible orders. How many words are possible, if all these words are written as in a dictionary? What will be the rank of the word ZENITH?"

What would be the solution and how should I approach or think to solve such permutation problems?

Answer 1

Let us calculate all the words that rank higher than “ZENITH”.

1. All words with any letter but “Z” as the first letter ranks higher than “ZENITH”. There are $5\times 5!$ words.
2. All words with “ZE” as first two letters and “I” or “H” for the third letter ranks higher than “ZENITH”. There are $2\times 3!$ words.
3. All words with “ZEN” as first three letters and “H” as the fourth letter ranks higher than “ZENITH”. There are $1\times2!$ words. 4
4. ”ZENIHT” rank higher than “ZENITH”. Another word.

Since there are $5\times 5!+2\times 3!+2!+1$ words that ranks higher than “ZENITH”, “ZENITH” itself ranks at $5\times 5!+2\times 3!+2!+2$.