All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is _________.

Solution

Maths sir · 4 steps

Fix letters from the left and count every word that comes earlier, keeping in mind there are two T's.

1. Arrange the letters. The 7 letters in alphabetical order are E, G, N, T, T, W, Y. The letter T appears twice, so any group of n letters that still contains both T's gives n!/2! arrangements, not n!.
2. Count words before it. Go letter by letter. At each position, count the words that match GTWENTY up to that point but have a smaller letter in that position.

   Words starting with	Letters left to arrange	Count
   E	G, N, T, T, W, Y	6!/2! = 360
   GE	N, T, T, W, Y	5!/2! = 60
   GN	E, T, T, W, Y	5!/2! = 60
   GTE, GTN, GTT	4 distinct letters each	3 × 4! = 72
   GTW	E, N, T, Y	GTWENTY is the first word here

3. Check the last letters. After GTW the letters left are E, N, T, Y, and GTWENTY uses them in exactly that order. So no other word starting with GTW comes before it.
4. So. There are 360 + 60 + 60 + 72 = 552 words before it, so GTWENTY is word number 552 + 1 = 553.

The common slip: 1033. Forgetting to divide by 2! for the two T's in the first three rows gives 720 + 120 + 120 + 72 + 1 = 1033, not 553.

The idea to keep. For rank questions, count all words that come before yours, then add 1. Each time, check if a repeated letter is still in the leftover group, and divide by its factorial only if it is.

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