So you need to reject that partial payment.http://www.landlordguidance.com/eviction-notice-fo...Tenant

I had one tenant that was rejected, and another that was accepted, but for the life of me I couldn't figure out why! 

Now the counter-offer is the offer and you can accept or reject it.So, when you verbally offered to renew the lease and he came back with - paint the apartment the way I want and I'll stay - that was a counter-offer. 

If they fail your application process you have a little less obligation - although you may have a legal one if it is a credit-based rejection.

So, when you can do targeting advertising, just on the law school housing page, or maybe on the hospital's facebook page or bulletin boards, you have a lot less rejecting to do.

They also reject to pay the rent if I do not fix the basement right away.

@Dan Shelhamer, you must check your county/state/city laws because some areas indeed do consider it discrimination to purposely reject people just for being on Section 8.

So, if you can legally reject one of these applicants, do yourself a favor and do so.

They might reject the lease offer, but it helps you. 

Credit reports are a extremely important factor but by no means the total picture.The purpose of the screening process is to identify reasons to reject a applicant.