The letters of the word KANPUR are written in all possible orders, and these words are written in dictionary (alphabetical) order. The word that appears at the $440^{\text{th}}$ position is
Step-by-Step Solution
Key Concept: Fix each letter position in alphabetical order and count $5!$, $4!$, $3!$ blocks to narrow down to the exact 440th word.
Sorted letters: A, K, N, P, R, U.
A…: $5!=120$ (positions 1–120).
K…: $5!=120$ (positions 121–240).
N…: $5!=120$ (positions 241–360).
PA…: $4!=24$ (positions 361–384).
PK…: $4!=24$ (positions 385–408).
PN…: $4!=24$ (positions 409–432).
PRA…: $3!=6$ (positions 433–438).
PRKA…: remaining letters N, U in order: PRKA**N**U (439), PRKA**U**N (440).
The $440^{\text{th}}$ word is **PRKAUN**.
Correct Answer: 3