How many strictly increasing functions $[4] \to [12]$ precede $(2,3,4,5)$?
Notice that out of the tuples that start with $2$. one has that $(2,3,4,5)$ is the smallest one. So you want to count the ones that start with $1$ and that would be all of them, so choose out of $13$ numbers the $3$ you will place after $1$ say $x_2,x_3,x_4$ such that $1<x_2<x_3<x_4$ in $\binom{13}{3}$ ways.