jmc

Since $15=3^1\cdot 5^1$, the largest possible value of $n$ for which $15^n\mid 942!$ is the largest possible value of $n$ for which both $3^n\mid 942!$ and $5^n\mid 942!$. Since $942!$ has many more factors of 3 than it does 5, our answer will be the number of factors of 5 in $942!$. $$\frac{942}{5}=188\frac{2}{5}\frac{188}{5}=37\frac{3}{5}\frac{37}{5}=7\frac{2}{5}\frac{7}{5}=1\frac{2}{5}$$There are $188+37+7+1=233$ factors of 5 in $942!$, so the largest possible value of $n$ is $\boxed{233}$.