AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_24 / How many strings of length 5 formed from the digits 0, 1, 2, 3, 4 are there such that for each j∈{1,2,3,4}, at least j of the digits are less than j? (For example, 02214 satisfies …

How many strings of length 5 formed from the digits 0, 1, 2, 3, 4 are there such that for each j∈{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.)

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 $2$.)