Is it permissible, Islamically, for a sister to study with her male colleague (who is not a Mahram) in her private apartment, in the presence of the professor, in order to complete the required number for the professor's teaching?

It is not permissible for the sister and her female colleague to study in an apartment in the presence of the teacher and another person, as this may lead to fitna (temptation or discord), given that Islamic law emphasizes strictness regarding men entering upon women. It has been reported in a hadith that the Prophet, peace and blessings be upon him, said: "Beware of entering upon women."

And based on the hadith narrated by Muslim: "No man shall enter upon a woman whose husband is absent from this day forth, unless there is with him one or two men," it can be interpreted as referring to a righteous group from whom the occurrence of immorality is averted. Therefore, if there is a genuine need to study with this man, and the controls mentioned by scholars are met, and fitna is safeguarded against, then it is permissible for them; otherwise, it is forbidden. It is preferable for this lesson to be received in a safe institute or educational institution.

As for the hadith: "Seek knowledge even if it be in China," its acceptance varies among scholars due to the weakness of its chains of narration. Even if we assume its authenticity, deriving a ruling from it in this context is incorrect. Furthermore, one must beware of issuing fatwas (religious edicts) without knowledge, as there is a severe warning against it.