The correct answer is option 3, 1664. The French East India Company was founded in 1664 by Jean-Baptiste Colbert, the finance minister of King Louis XIV of France. It was established as a government-controlled monopoly for trade between France and the East Indies, which included regions such as India, China, and Southeast Asia.

Option 1, 1600, is incorrect as it predates the actual founding of the French East India Company. Option 2, 1620, is also incorrect as it falls before the actual establishment of the company. Option 4, 1604, is incorrect as it is too early and does not correspond to the actual founding year of the company.

Therefore, the correct answer is option 3, 1664.