If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315ᵗʰ position in this arrangement is :

Solution

Maths sir · 4 steps

Fill the dictionary block by block: count whole blocks until you are close to 315, then list the last few.

1. Order the letters. The six letters are all different. In dictionary order they are A, G, N, P, R, U. Fixing the first letter leaves 5! = 120 words, and fixing the first two leaves 4! = 24.

   Words starting with	How many	Positions covered
   A	5! = 120	1 to 120
   G	5! = 120	121 to 240
   NA	4! = 24	241 to 264
   NG	4! = 24	265 to 288
   NP	4! = 24	289 to 312
   NR	4! = 24	313 to 336

2. Enter the NR block. The NR block starts at position 313, so the word you want is the 3rd word in this block. The letters left are A, G, P, U, so the block begins with NRA followed by G, P, U in dictionary order.
3. List the first three. Position 313 is NRAGPU, 314 is NRAGUP and 315 is NRAPGU.
4. So. The word at position 315 is NRAPGU, which is option (c).

The idea to keep. For a word at a given position, jump in whole blocks of (n-1)!, then (n-2)!, and so on. Only list words one by one when fewer than a handful are left to count.

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