My mom gave someone in India access to her computer

Question

I woke up to a frantic phone call from my mom this morning. As far as I could put together, this was the timeline of events that she went through:
1) She was having trouble typing emails in her Yahoo mail account via Firefox. Where most of us would probably update the software, she did a search for help with this problem via Google.
2) She somehow found a phone number that was claiming to be Yahoo tech support. She called it and the person on the other line tells her she has been hacked, throwing my mother into a panic. The scammer on the other end (which I'm convinced they were) has her somehow download a file, I'm assuming via email and just GIVES THIS PERSON ACCESS TO HER COMPUTER!!!
3) The person now has remote access to the computer and is controlling her desktop. She tells her via some sort of chat window (I think, my mom was very unclear on how this information was coming to her... like I said she was panicked) that she has been hacked and needs to call some number to pay $250 to fix the problem. Of course, she would have to give this person her credit card info.
4) At this point, she calls me (I'm deeply asleep and taking in most of this information through a half-awake haze). She tells me what's going on, I find out the number she called and google it to find it goes to something called "AGILE" and another called "Techies365.com". I tell her to hang up on these people RIGHT NOW and shut down her computer and unplug it from the internet.
SO... As far as I can tell, it seems like this is one giant scam where they take advantage of clueless people like this, convince them they've been hacked and then extort them for $250.
My question is what does she need to do to make sure she uninstalls anything that these a-holes have installed on her machine? She didn't give them any passwords other than typing in her administrator password, but I don't know what else they managed to access on her computer. She unfortunately doesn't have a Time Machine backup she can restore from.

Answer 1

This is a well-known scam. I haven't read of any kind of malware being installed by the scammers, but there's no way to be sure that didn't happen. She should back up her data, then erase the startup volume, reinstall OS X, and restore only her user data and settings from the backup. Third-party software must be reinstalled from scratch.
Then she should change every Internet password she has and check all her financial accounts for unauthorized transactions. If she actually did give out her credit card number, the card has to be canceled immediately.