JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Permutations and Combinations

A number is divisible by $3$ when its digit sum is. Among $1,3,5,8$ the residues modulo $3$ are $1,0,2,2$. Counting three-digit strings whose digit sum is a multiple of $3$: the equal-digit numbers $111,333,555,888$ give $4$; the multiset $\{5,5,8\}$ (sum $18$) gives $3$ arrangements and $\{8,8,5\}$ (sum $21$) gives $3$; the all-different sets $\{1,3,5\}$ and $\{1,3,8\}$ give $3!=6$ arrangements each. The total is $4+3+3+6+6=22$.