How many free copies of a textbook is a publisher typically willing to provide? Normally I teach a single section with 10-15 students, and a single copy is sufficient.  This class will have 3 sections of about 20 students each.  FWIW, I am in the United States.

I am considering changing textbooks for an introductory class.  A problem is that none of the TAs and tutors already have this textbook, and our department funds are limited.  

How many free desk copies is a publisher usually willing to provide?
You have to understand that in the undergraduate textbook market, the publishers are competing intensely for textbook adoptions and will do what it takes to make the sale. (This is not so much the case for e.g. the monographs that might be used in graduate courses). The marginal cost of production of an additional book is very low, people who get desk copies were quite unlikely to purchase for their own use, and so it costs the publisher very little to give out desk copies ad libitum. 

At my large R1 school we have different TAs for our intro classes every quarter; sometimes ten or more of them. We have no problem getting desk copies for each TA.