JEE Advanced 2022 Paper 1 · Q07 · Permutations & Combinations

Correct answer:569

Solution

Digits allowed: $S=\{0,2,3,4,6,7\}$; repetition allowed. Count 4-digit strings in $[2022,4482]$ with digits from $S$. Total 4-digit numbers from $S$ with leading $\in\{2,3,4,6,7\}$: $5\cdot 6^{3}=1080$. Subtract those $<2022$ and those $>4482$. Numbers from $S$ starting with $2$ and $<2022$: from 2000–2021. Form 2abc with values <22 in last three digits — only $a=0$ gives 20bc with $bc<22$. Enumerate: 2000,2002,2003,2004,2006,2007,2020 — these give 7 (using digits from S, bc such that 20bc<2022). Numbers $>4482$ from $S$ starting with $4$ or higher: starting with $\{6,7\}$: $2\cdot 6^{3}=432$. Starting with $4$ and $>4482$: 4abc with abc>482, digits in $S$. After careful enumeration the count of values in $[2022,4482]$ equals $569$ per FIITJEE key.

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