HMMT November 2024 E4

Compute the number of ways to pick a 3 -element subset of

$$\left\{10^{1}+1,10^{2}+1,10^{3}+1,10^{4}+1,10^{5}+1,10^{6}+1,10^{7}+1\right\}$$

such that the product of the 3 numbers in the subset has no digits besides 0 and 1 when written in base 10 .