Which of the following rulers gave permission to the East India Company to…

European trading companies in Mughal India needed an explicit royal permission — a farman granted by the ruling Mughal emperor — before they could set up factories and trade freely in the empire's territory. Such a farman was issued by whichever emperor held the throne when a company's ambassador successfully petitioned the Mughal court.

The English East India Company, chartered in 1600, first gained a toehold at Surat in 1613. In 1615, King James I sent Sir Thomas Roe as ambassador to the Mughal court, then held by Emperor Jahangir. After prolonged negotiations at Jahangir's court (1615–1619), Jahangir issued a farman granting the Company the right to build factories and trade across his dominions, beginning with Surat — this is the formal permission to trade referred to in the question.

• Bahadur Shah Zafar was the last Mughal emperor, reigning 1837–1857 — by his time the Company already controlled most of India as a colonial power, well over two centuries after the actual trading permission was granted.

• Humayun reigned 1530–1540 and 1555–1556 — decades before the English East India Company was even chartered in 1600, so no permission episode involving the Company could have occurred in his reign.

• Akbar reigned 1556–1605 — no English mission secured a farman for formal trading rights during his rule.

Hence, it is Jahangir who granted the East India Company permission to trade in India.