JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Permutations and Combinations

A number is divisible by $6$ when it is even with digit sum divisible by $3$, so its units digit is $2$ or $4$. Modulo $3$ the available digits split as $\{3\}$ of residue $0$, $\{1,4\}$ of residue $1$ and $\{2,5\}$ of residue $2$. Counting ordered pairs $(a,b)$ for the first two places, the digit sum $a+b$ is $\equiv 0,1,2(\mathrm{mod}3)$ in $9,8,8$ ways respectively. If the units digit is $2$, the pair needs $a+b\equiv 1(\mathrm{mod}3)$, giving $8$ numbers; if it is $4$, it needs $a+b\equiv 2(\mathrm{mod}3)$, giving $8$ numbers. Therefore the total number of such three-digit numbers is $8+8=16$.