The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

Solution

Maths sir · 3 steps

Fix TOUGH one letter at a time, count the words that start with a smaller letter, then add 1.

1. Alphabetical order. The letters of OUGHT in dictionary order are G, H, O, T, U. All five are different, so 5! = 120 words in all.

   Fixed so far	Smaller letters available	Words skipped
   first letter T	G, H, O	3 × 4! = 72
   T, second letter O	G, H (from G, H, O, U)	2 × 3! = 12
   TO, third letter U	G, H (from G, H, U)	2 × 2! = 4
   TOU, fourth letter G	none (from G, H)	0 × 1! = 0

2. Add them up. There are 72 + 12 + 4 + 0 = 88 words before TOUGH in the dictionary.
3. So. TOUGH comes right after these 88 words, so its serial number is 88 + 1 = 89, which is option (c).

The idea to keep. For the rank of a word, at each position count the unused letters smaller than the one placed there, multiply by the factorial of the places left, add everything, and finally add 1 for the word itself.

Answer from the NTA final answer key. Solution by PrepWiser.