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Is it common phenomenon to invite partners and colleagues over dinner at your house in Mumbai ?

Does it happen or its non existent in law firms
That might sound trollish, but is perhaps a good point. In principle, having dinner with your colleagues and partners is fine, but you have to be in a position where you are confident enough to judge the social and professional dynamics in the workplace to whether it would be welcome or awkward, and that is hard for someone else who is not in your shoes, to do.

Some factors to consider: how long have you been at a firm, do you ever talk about family or outside-work topics with them, have you been out for coffee or lunch with them socially already, etc...
I agree, and these two are genuinely good comments. I've never thought of it that way, but now that I look at my professional relationships that have developed stronger/more personal, yeah, this is actually the correct approach.
Inviting them home for dinner: ok

Sleeping with them after that: not ok
It will be awkward unless you have worked with them for a few years. Better to organise it as birthday lunch or dinner at a nearby restaurant from work.