Question

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 315^th position in this arrangement is

• A. NRAGUP
• B. NRAGPU
• C. NRAPGU
• D. NRAPUG

Answer 1

The Correct Option is C

Arranging the letters in alphabetical order: NAGPUR

Starting with \( A \): \( 5! = 120 \) positions

Starting with \( G \): \( 5! = 120 \) positions, cumulative: 240

Starting with \( N \) and \( A \): \( 4! = 24 \) positions, cumulative: 264

Starting with \( N \) and \( G \): \( 4! = 24 \) positions, cumulative: 288

Starting with \( N \) and \( P \): \( 4! = 24 \) positions, cumulative: 312

Now, starting with \( N \), \( R \), and \( A \):

\[ \text{NRAGUP} = 1, \text{ cumulative: 313} \]

\[ \text{NRAGPU} = 1, \text{ cumulative: 314} \]

\[ \text{NRAPGU} = 1, \text{ cumulative: 315} \]

Thus, the word at the \( 315^{\text{th}} \) position is NRAPGU.

Answer 2

To determine the 315^th word in the dictionary arrangement of the letters of the word "NAGPUR", we follow these steps:

• The word "NAGPUR" consists of 6 different letters: N, A, G, P, U, and R.
• In a dictionary arrangement, words are alphabetically ordered. Thus, we first list the letters alphabetically: A, G, N, P, R, U.

We calculate the total number of permutations starting with each letter until we reach the desired position:

1. Start with letter A:
   • Words starting with A: \(5! = 120\) words (since 5 letters are left: G, N, P, R, U).
2. Start with letter G:
   • Words starting with G: \(5! = 120\) words.
3. Start with letter N:
   • Words starting with N: We exhaust combinations starting with NA, NG, and finally reach combinations starting with NP.
     • Starting with NA: \(4! = 24\) words.
     • Starting with NG: \(4! = 24\) words.
     • Starting with NP:

Now, calculate from NP to find position 315:

• Next sequence starting with NP, we try letter:
  • NR: \(3! = 6\) words starting with NR.
  • Start with NPU: Skip as 6 positions are exhausted by NR already.
  • Consider NR:
    • NR places at position \(240 + 48 = 288\) (120 + 120 + 24 + 24).
    • Additional permutations needed: \(315 - 288 = 27\) after NR.

Continuing with NRA (R can be followed by A, G, P, U forming 4! each):

• NRAGU: \(3! = 6\) permutations, omitted.
• NRAP: (NRAPG, NRAPU)
  • This brings us close to: NRAPGU - the word at position 315^th.

Thus, the word at the 315^th position is NRAPGU.