Which factor european traders gain a foothold in the once-powerful mughal empire in the 1600s?

The answer you are looking for is option 3. Heavy taxes imposed by Aurangzeb.

Nurjahan, the father of Aurangzeb, began to build the Taj Majal that was being built for his deceased wife. This implied that he was not doing things for the people of the empire.

In the late 1600s, Emperor Aurangzeb rejected the tolerant policies of Akbar and resumed the persecution of the Hindus that existed before him, which caused economic difficulties increased under heavy taxes and discontent sparked revolts against the Mughal rule.

These conditions favoured European merchants to take hold in the powerful Mughal Empire.

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