  Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$
or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}:
n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a
complete system of residues modulo any power of p.
