CUET PG 2024 — Mathematics PYQ

$\text{Word: AGAIN}$

$\text{Letters: }A,A,G,I,N$

$\text{Total number of distinct permutations}<br>=\frac{5!}{2!}=60$

$\text{Step 1: Fix the first letter}$

$\text{Words starting with }A=4!=24(\text{positions }1\text{ to }24)$

$\text{Words starting with }G=\frac{4!}{2!}=12(\text{positions }25\text{ to }36)$

$\text{Words starting with }I=\frac{4!}{2!}=12(\text{positions }37\text{ to }48)$

$\text{Words starting with }N=\frac{4!}{2!}=12(\text{positions }49\text{ to }60)$

$\text{Hence, the }49^{\text{th}}\text{ word is the first word starting with }N.$

$\text{Step 2: Arrange the remaining letters}$

$\text{Remaining letters: }A,A,G,I$

$\text{Lexicographically smallest arrangement: }AAGI$

$\text{Therefore, the }49^{\text{th}}\text{word is }\boxed{\text{NAAGI}}$