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MBB's mostly hire Engineers and B-school folks. Some freshers from commerce, economics, finance-related degrees also get in.

I'd like to know how many law students are hired by these companies? I understand that it's difficult - but is it possible through effective networking on LinkedIn? Any idea on their hiring cycle for non-target streams such as law?

Moving this up because of no answers :/

Koi toh kuch bolo!
Lawyers are mostly useless on aspects other than regulatory advice/ contract vetting. You need to be really strong in maths/ statistics/ eco to develop models, etc. to be relevant in consulting, not a lawyer's forte.
Never in the consulting arm. I have seen 1 or 2, but with proper jugaad.
McK hires from Law School. I think once they hired from NALSAR (or shortlisted someone from NALSAR). BCG/Bain do not. In any case, they hire for the BA role, not Associate (for which an MBA is almost mandatory).
McK has stopped hiring from NLS (& any other Indian law school) directly for quite some years now.
Maths sikhate hain "law school" mein? Agar nahi toh shayad chai banwate honge baccho ko recruit karke
MBB do not hire actively from law schools anymore. McK used to go to NLS quite regularly, but don't think they do anymore. Last I know, McK went to NLS and NUJS in 2019, interviewed a few people but did not make any offers. BCG went to NLS in the late 2000s but to my knowledge, not after that. This is primarily due to a couple of reasons: a) McK goes to hire usually only towards the second half of the fifth year and by that time most people already have offers from other places. They specifically look for people with stellar CVs which becomes tricky as those people have already accepted job offers elsewhere and are reluctant to apply that late, and b) Lawyers are regularly going to do MBAs from IIMs now, so they can hire them from there more easily.
If your target is MBB (or consulting in India) straight after law school then you should either consider doing an MBA from one of the top IIMs or look at the Young India Fellowship by Ashoka.
Doing it via networking is very difficult as these firms literally have thousands of people applying every year. You should be persistent, have a stellar CV, and/or know someone there to have a shot.
Nice. This is helpful.

Coming to a stellar CV - definitely trying hard to make one. Have been working for a startup in a Business role alongside college and have also worked a consultant before. Plus, the usual law firm internships are on. Keeping my eyes open and networking alongside :)
A Law Schoolite a while ago joined McK straight out of college, and was even accelerated to permanent track in NYC (without the MBA), they stuck around till they were made EM and then moved laterally to a startup that then went public. But more recent Law School McK alum all moved out at the BA level itself.
You can't just make a stellar CV. It requires years of preparation.

They even look at school grades and need you to have held positions of responsibility. It is quite a trip.
I have joined Bain from law school. Sorry, I meant Bane. I joined a law firm.
Big consulting firms such as McKinsey and BCG hire fresh law graduates and even provide opportunities for internships as well. You can make good networks via LinkedIn and also use the platform to look up vacancies. The best approach is to look up vacancies on the respective career pages and keep applying based on your skill set. It is not rocket science. All the best!