I'm trying to prove that the following language is irregular using the Myhill-Nerode theorem
$$ L = \{ w\space\epsilon \{a,b,c\}^* | \#_b(w) > (\#_a(w) + \#_c(w))! \} $$
While it's completely clear that it's irregular using basic logic (the fact a regular expression can't "remember") and using the pumping lemma, I am completely stuck trying to show it with the theorem.
I've tried dividing up the: $$ (\#_a(w) + \#_c(w))! \to (\#_a(w) + \#_c(w))*(\#_a(w) + \#_c(w)-1)! $$ I've tried creating sub groups by: $$ \frac{(\#_b(w))}{(\#_a(w) + \#_c(w))} $$ But all to no avail.
Suggestions and/or help would be greatly appreciated :)