AMC 12 Mock Exam 1 Problem 3

We count three-digit strings with first digit nonzero and digit product $36$; this is the same as counting three-digit positive integers whose digits have product $36$. No digit is $0$, since the product is nonzero. The multisets of three digits from $1$ to $9$ with product $36$ are $\{1,4,9\}$, $\{1,6,6\}$, $\{2,2,9\}$, $\{2,3,6\}$, $\{3,3,4\}$. They can be arranged in $6$, $3$, $3$, $6$, and $3$ ways respectively, for a total of $6+3+3+6+3=\boxed{21}$.