Such a long reply repeating the same thing is very obnoxious. Don't do it. If I could downvote you I would. Do you really know what you're talking about, huh?

Such a long reply repeating the same thing is very obnoxious. Don't do it. If I could downvote you I would. Do you really know what you're talking about, huh? Your pigeon-hole example is pretty wrong because you can't use intuition (with completely wrong numbers!) to judge, you use math. Let's say there's a trillion pigeons and the hash we're using is 256 bits. That 2^256 ~ 10^77. Divide that by a trillion and you still have 10^65. Vastly more than enough to be secure all the while solving two problems. First that with a specific user's password one can often access other services. Two, and most importantly, that the password database blows up and isn't feasible to steal any more, either over internet or in real life.