(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to sum the binomial coefficients that are divisible by a

positive integer t, i.e.

[tex]\sum_{i=0}^{s}\binom{ts}{ti}[/tex]

Is there any way to get rid of the sum sign?

2. Relevant equations

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

infinity?

[tex]EX=\frac{\sum_{i=0}^{s}ti\binom{ts}{ti}}{\sum_{i=0}^{s}\binom{ts}{ti}}[/tex]

3. 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.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Divisible binomial coefficients

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