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
- NRAGUP
- NRAGPU
- NRAPGU
- NRAPUG
The Correct Option is C
Approach Solution - 1
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.
Approach Solution -2
To determine the 315th 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:
- Start with letter A:
- Words starting with A: \(5! = 120\) words (since 5 letters are left: G, N, P, R, U).
- Start with letter G:
- Words starting with G: \(5! = 120\) words.
- 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:
- Words starting with N: We exhaust combinations starting with NA, NG, and finally reach combinations 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 315th.
Thus, the word at the 315th position is NRAPGU.
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