Problem 10 · Multiples of 4 with digit sum 4

Reasoned solution

Key idea: a number is a multiple of $4$ according to its last two digits, and those must use up little of the sum.

The two-digit endings that are multiples of $4$, have digit sum $\le 4$ and leave room for the leading digit ($\ge 1$) are $00$ (sum $0$), $20$ (sum $2$) and $12$ (sum $3$). (With $04$ or $40$ the remaining digits would sum $0$ and the leading digit couldn't be $\ge 1$.)

Ending $00$: the first three digits sum to $4$ with the first $\ge 1$: $\binom{5}{2} = 10$ ways. Ending $20$: the first three sum to $2$: $110$, $101$, $200$, i.e. $3$ ways. Ending $12$: the first three sum to $1$: only $100$, $1$ way.