Here's how Tilly got in...
She's the only kid who showed up, becuase every one else had the flu, there was a power cut, and some fucker declared war on the klingons.
Last woman standing because no other woman was ever standing near her.
(No conformation on that, but other than wearing a really a short dress, even shorter than normal in TOS, I do not know how this woman got into the academy with her defectivities on display. Hitler, Khan, and Colonel Green were all in a position to stamp out gingers, and failed.)
Although a smart applicant can game the system, doing on purpose what Tilly did by accident, by taking their exam in a bad area full of dummies, or an area with a low to zero turn out.