All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :

Solution

Maths sir · 3 steps

Go letter by letter and count every word that would sit before PUBLIC in the dictionary.

1. Alphabetical order. The six letters are all different. In dictionary order they are B, C, I, L, P, U.
2. Counting rule. At each position, see how many unused letters are smaller than the letter PUBLIC actually has there. Each of them can start a block of earlier words, and each block has (letters left after it)! words.

   Position	Letter in PUBLIC	Unused smaller letters	Words placed before
   1st	P	B, C, I, L (4)	4 × 5! = 480
   2nd	U	B, C, I, L (4)	4 × 4! = 96
   3rd	B	none	0 × 3! = 0
   4th	L	C, I (2)	2 × 2! = 4
   5th	I	C (1)	1 × 1! = 1
   6th	C	none	0

   480 + 96 + 0 + 4 + 1 + 0 = 581

3. So. Exactly 581 words come before PUBLIC, so PUBLIC itself is word number 581 + 1 = 582, option (d).

The idea to keep. For the rank of a word, count all the words that come before it, one position at a time, then add 1 for the word itself. Once a letter is used, never count it again.

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