The correct answer is option 2, Qasim Bazar. In 1651, the Mughals gave permission to the East India Company to trade and establish a factory in Qasim Bazar, which is located in Bengal. This permission allowed the East India Company to carry out its trading activities in the region and build a factory to support their operations.

Option 1, Calcutta, is incorrect as the East India Company received permission to build a factory in Calcutta in 1690, not 1651. Alert - The correct answer should be Calcutta, not Qasim Bazar.

Option 3, Singur, and option 4, Burdwan, are also incorrect as these places were not mentioned in the context of granting permission to the East India Company for trade and building a factory by the Mughals in 1651.