Must I Run Credit/Background check on ALL applicants?

[Hans M.]

Newbie here. It's my first time screening tenants, and I'm wondering if it's okay to ask for background/credit reports for only the most qualified applicants, according to the information they filled out on my rental application. So for example, seven people fill out a rental application, but I only ask for the credit/background report on the top three out of those seven applications, based on what they report as their income, credit score, etc. My thought was that this would save time/money for the less qualified potential tenants, as well as encourage more people to apply, as opposed to look at the application fee ($30 per person), balk, and walk away.

Are there any potential problems with this method?