KöMaL 2023 C1767

Given are $2$ coins of $7$, $3$ coins of $17$, $5$ coins of $119$, $7$ coins of $289$, $11$ coins of $2023$ and $n$ coins of $1$. Two coins are selected at random, and their values are multiplied together, and a result of $2023$ is obtained. Find the value of $n$, given that the probability of obtaining that result is $\frac{12}{55}$.