What is the ruling on a male student shaking hands with his female classmate at university, and what should he do if she extends her hand to greet him?

Mixed-gender education is impermissible due to the potential for fitna (temptation/strife) therein. Similarly, it is not permissible to shake hands with a non-mahram woman, based on the Prophet Muhammad’s (peace be upon him) saying: "Indeed, I do not shake hands with women," and Aisha's (may Allah be pleased with her) statement: "By Allah, the hand of the Messenger of Allah (peace be upon him) never touched the hand of any woman, ever." Handshaking is one of the means of fitna for both parties. As for the permissible Islamic greeting, free from fitna, without handshaking, suspicion, or a submissive tone of voice, and with hijab and the absence of khalwa (private seclusion), there is no harm in it. It is permissible for a woman to shake hands with other women and with her mahram men.