Yes you can choose any centre across the country as per your convinience. The allotment of the centres will be done on the basis of availability of seats in the centre. If the exam you are appearing for, is not conducted from the city of your choice then,you will have to opt some other centre.
Yes exam centres can be chosen across India. But you have to check if there is exam centre available in Mumbai or not. If you provide your choice of exam centre in Mumbai you will get it but it is again subject to availability.
If you clarify for which exam you are asking this question then we can provide more clarity on the same.
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