For p a constant in (0,1) and n going to infinity such that pn is an integer, consider the distribution on n bits that selects a random subset of pn bits, sets those to 1, and sets the others to 0. What is the largest k = k(n,p) so that the induced distribution on any k bits is 1/10 close in total variation distance (a.k.a. statistical distance) to the distribution that sets each bit to 1 independently with probability p? For every p I would like to know k up to a sublinear (i.e. o(n)) additive term. (For starters, p = 1/8 is good too.) Does anybody know of a place where this is worked out?

Thanks! Emanuele