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This may seem like an obvious question to ask, but I haven't been able to find a definite answer anywhere: do I ask my undergrad institution to send one copy of my transcript to the LSAC (who will then copy it and send it out to the 11 schools I am applying to) OR do I request 11 copies of the same transcript to send to the LSAC?

One copy. LSAC will copy, summarize, and forward all the necessary transcript info for each of the schools you apply to.