If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

Solution

Maths sir · 3 steps

Count every word that comes before THAMS, letter by letter, then add 1 for THAMS itself.

1. Alphabetical order. Arrange the letters as a dictionary would: A, H, M, S, T. All five letters are different, so there are 5! = 120 words in total.
2. Skip earlier blocks. At each position, count the unused letters that come before the letter of THAMS. Each such letter starts a block of words that comes earlier, and the block size is the factorial of the number of places left after it.

   Position	Letter in THAMS	Unused letters before it	Words skipped
   1st	T	A, H, M, S (4 letters)	4 × 4! = 96
   2nd	H	A (1 letter)	1 × 3! = 6
   3rd	A	none	0
   4th	M	none (only M, S left)	0
   5th	S	none	0

3. So. There are 96 + 6 = 102 words before THAMS, so THAMS has serial number 102 + 1 = 103.

The idea to keep. Rank of a word = (number of words that come before it) + 1. At each position, multiply how many smaller unused letters there are by the factorial of the places left.

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