Which Buddhist Council took place during the reign of King Ashoka?

Third

The correct answer is Option 3- Third

The Third Buddhist council was convened in about 250 BCE at Asokarama in Pataliputra, under the patronage of Emperor Ashoka.

Buddhist Councils: Location and Presiding Figures

First Council:
Location: Sattapanni Caves, Rajgriha (present-day Rajgir)
Patronage: King Ajatashatru
Presided by: Mahakassapa (monk)

Second Council:
Location: Vaishali
Patronage: Uncertain
Presided by: Uncertain (possibly several elders)

Third Council:
Location: Pataliputra (present-day Patna)
Patronage: Emperor Ashoka
Presided by: Moggaliputta Tissa (monk)

Fourth Council:
Location: Kundalvana, Kashmir
Patronage: King Kanishka
Presided by: Vasumitra and Ashvaghosha (monks)