Calculate the following number
$$\sup \{x\in[0, 1] \space | \space \#A_x = \infty \}$$
Where $A_x = \{n\in \mathbb{N} \space | \space \text{the fraction of $n$'s digit permutations that are prime } \geq x\}$.
The digit permutations of a number are all the permutations of its digits, for example for $n=2399$ the permutations are
$$\{ 2399, 2939, 2993, 3299, 3929, 3992, 9239, 9293, 9329, 9392, 9923, 9932 \}.$$
Seven out of these twelve numbers are prime: $\{2399, 2939, 3299, 3929, 9239, 9293, 9923\}$ so $n\in A_{\frac{7}{12}}$.
