I need to sum the binomial coefficients that are divisible by a
positive integer t, i.e.
Is there any way to get rid of the sum sign?
Let t be fixed and s go to (positive) infinity (both t and s are
positive integers). Let M(s) be a set with #M(s)=ts, then I am really
interested in the expected value of the number of elements when you
choose subsets from M whose cardinality is a multiple of t. For
example, what is the mean number of elements picking subsets with
cardinality 0, 3, 6, or 9 from a set with cardinality 9 (t=3, s=3)?
Where does this expected value go as s (the ``grain'' of M) goes to
The Attempt at a Solution
I anticipate the solution to be lim(s->infty)EX(s)=ts/2, but I'd love
to prove it.