rush

^ Not quite.
There's a quota for bids that they will give out. And it is the number of girls preffed/# of houses.
When you hear of girls not getting bids, it is because they are released early, or because they were crosscut by houses. Therefore, not all houses meet quota, and some houses give out snap bids (bids to girls that were cut from recruitment given last minute to try and reach quota).
Basically, the more girls that go through recruitment, the more larger the PC.

so a pnm's chances of receiving a bid from XYZ house aren't decreased due to large # of overall girls?

To not be cross cut/released early, I would definitely make sure your appearance is great. By that I mean presenting the best possible version of yourself and being very likable to every girl at every house, even if you may not see yourself at that house immediately. Second and most important, be realistic. If you start to feel like your chances at a house are very slim or know that you probably won't fit in even if you really love it, the members are probably thinking the same thing, so concentrate more on other houses that you get a good vibe from and that you can see yourself in. I'm not saying you should sell yourself short or not be confident, just be aware of where you feel like you fit in best!

Hey!
To avoid being crosscut:
Crosscut means that you didn't rate the houses that wanted you high enough and the houses that you wanted didn't rate you high enough. Basically, you are not put on a bid list because sometimes the system is a little faulty. To avoid this, put every house on your preference card after prefs round. If you preffed at ABC, DEF, and GHI, put all three on the card. If you preffed at ABC and DEF, put those two on the card, and if you only pref at one, put it on your card. Obviously rank them, but put them all down, even if you hate GHI. This is because if you are crosscut, the computer Pi Chi goes in and manually sorts you into a house as long as you maximized your options and put all of these houses on the card. If you are crosscut, and did not maximize your options, you will be dropped and not manually placed in a house.
To avoid being released:
When you are released, you are not invited back to any house, anytime during recruitment. You could be released after Round 1, or you could be released after Round 3. To avoid this, dress well (but within your style), make sure to spend time and effort on your appearance (hair, makeup, body odor, etc.), and try and have geunine conversations at each house and get to know the girls. Also, do NOT skip a party. You will be released.

Best of luck. :)