The usual cited figure is six arrows a minute in actual combat conditions. The length of the battle itself ranges somewhere between two and five hours depending on the source. Of course, the different witnesses to the battle may have different interpretations of when the battle began or ended. There's no telling exactly how many arrows were shot from these figures, though. We don't know exactly when the archers began their volleys, how many of them participated in each one, and how many arrows were available to them at the beginning of the engagement. What we do know is that they didn't necessarily join in hand to hand combat after running out of arrows (although some may have been running low on ammunition). That was a critical role of English archers in this period and would have been done at some point whether there were arrows left or not. None of our sources about the battle mention fears of running out of arrows, which inclines me to believe that it wasn't much of a factor.

"Rate of fire" is an anachronism when applied to archery. Bows did not "fire" arrows. "Fire" is a term applicable to firearms, not to archery.