What options do rejected applicants have if mistakenly told they were admitted to a program?
I've been putting off answering this because it's really a legal question, not an academic question, and I'm a computer scientist.  In my one business law course, I learned that the essence of a contract is an offer and an acceptance.  We have those.

If I were in this boat, I'd probably suck it up and apply elsewhere.  If I really-really wanted to go to the school that first accepted, then rejected me, I'd consider paying a lawyer a few hundred dollars to call the counsel for the university.  My lawyer would say something like, "My client is damaged blah-blah and has cause of action blah-blah.  However, we do not want to be unreasonable.  All we want you to do is honor the offer you made in writing and later confirmed verbally.  Um, and keep these circumstances confidential from anyone who could assign a grade to or otherwise evaluate my client."

There is a huge down side to doing that.  It is that the circumstances are quite likely to leak out.  Then one is facing professors who know how one was admitted.