His half-sister from his father has a half-sister from her mother. Is it permissible for him to marry her?

Praise be to Allah.

Yes, that is permissible. 

Shaykh Saalih al-Fawzaan said: 

If you have a half-sister from your father, then it is permissible for you to marry her half-sister from her mother, because there is no relationship between you and her half-sister from her mother that would prevent that. Allaah says (interpretation of the meaning):  

“All others are lawful”

[al-Nisa’ 4:24]

al-Muntaqa, 5/258. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) narrated that the Muslims were unanimously agreed that this kind of marriage is permissible. He said: It is permissible for a man’s half-sister from his mother to marry his half-brother from his father. This is unanimously agreed upon among the Muslims and there is no dispute concerning that. And Allaah knows best. 

From al-Fataawa al-Kubra, 3/163.