JEE Main 2025 Jan 29 Shift 2, Mathematics Q10: Permutation

JEE Main2025Jan 29, Shift 2Mathematics

If all the words with or without meaning made using all the letters of the word “KANPUR” are arranged as in a dictionary, then the word at $44 0^{th}$ position in this arrangement, is :

A
PRNAKU
B
PRKANU
C
PRKAUN
D
PRNAUK

Solution

Alphabetical order of letters: A, K, N, P, R, U.

Count words starting with each prefix until cumulative count reaches $440$ :

A____ : $5! = 120$ (running total $120$ )

K____ : $5! = 120$ ( $240$ )

N____ : $5! = 120$ ( $360$ )

PA___ : $4! = 24$ ( $384$ )

PK___ : $4! = 24$ ( $408$ )

PN___ : $4! = 24$ ( $432$ )

PRA__ : $3! = 6$ ( $438$ )

Next come the words starting with PRK in alphabetical order of the remaining letters A, N, U. The two arrangements with prefix PRKA are PRKANU (position $439$ ) and PRKAUN (position $440$ ).

$∴$ The $44 0^{th}$ word is PRKAUN.

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JEE Main 2025 Jan 29 Shift 2, Mathematics Q10: Permutation

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