JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Permutations and Combinations

List the numbers in increasing order and count how many precede $42923$. A leading digit smaller than $4$ can be $2$ or $3$ (not $0$), each followed by any four digits: $2\times 6^4=2592$ numbers. With first digit $4$ and a smaller second digit, only $0$ qualifies, giving $6^3=216$. With $42$ fixed and a third digit below $9$, the choices $0,2,3,4,7$ give $5\times 6^2=180$. With $429$ fixed and a fourth digit below $2$, only $0$ works, giving $6$. With $4292$ fixed and a fifth digit below $3$, the digits $0,2$ give $2$. The count preceding $42923$ is $2592+216+180+6+2=2996$, so its serial number is $2997$.