2022 AMC 10A Problem 24

How many strings of length $5$ formed from the digits $0$ , $1$ , $2$ , $3$ , $4$ are there such that for each $j \in \{1,2,3,4\}$ , at least $j$ of the digits are less than $j$ ? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$ , at least $2$ digits less than $2$ , at least $3$ digits less than $3$ , and at least $4$ digits less than $4$ . The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$ .)