Matrix permanent, #P-hard problems and NP
Let's take SAT, the quintessential NP-complete problem, as an example. An instance of SAT is a formula, and the answer is whether the formula is satisfiable. The corresponding #P-hard problem #SAT is instead to count the number of satisfying assignment. Clearly if you can count the number of satisfying assignments then you can tell whether one exists. In particular, if you can solve #SAT in polynomial time, you can solve SAT in polynomial time.

This shows that if some #P-hard problem can be computed in polynomial time, then all problems in NP can also be solved in polynomial time.