Management consulting firms will recruit students who come from all types of backgrounds. What's most important is that you show initiative. Also, understand that many consulting firms recruit primarily at their target schools, and it is difficult to be accepted if you don't attend one of these programs.
As an example, in terms of the big international firms that recruit heavily, many focus on these 7 programs.
It can be really difficult to land at McKinsey or BCG if you do not go to a school that they recruit at. This is why we recommend contacting HR and just asking - do you recruit at x school? Where do you recruit?
I hope this was of help.
Conrad and the Stacy Blackman
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MBA blogger, US News and Author, The MBA Application Roadmap