Register to reply

Divisible binomial coefficients

Share this thread:
stlukits
#1
Jul3-11, 11:30 AM
P: 39
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
Phys.Org News Partner Science news on Phys.org
Scientists discover RNA modifications in some unexpected places
Scientists discover tropical tree microbiome in Panama
'Squid skin' metamaterials project yields vivid color display
stlukits
#2
Jul10-11, 11:53 AM
P: 39
Clarifying and rephrasing:

I have two finite sets A_{1} and A_{2} with the same number of
elements (let their cardinality be s times t, where t is a fixed
positive integer). Let me randomly pick elements from these two sets,
with one constraint, however: the number of elements picked from A_{2}
must be a t-multiple of the number of elements picked from A_{1}. If
t=3, for example, and s=2, there are six elements in A_{i} and I can
either pick 0 elements from A_{1} and 0 from A_{2} (there is only one
way of doing this), or 1 element from A_{1} and 3 elements from A_{2}
(there are 120 ways of doing this), or 2 element from A_{1} and 6
elements from A_{2} (there are 15 ways of doing this). Let X be the
random variable counting the elements picked from both sets. In
the example, X can be 0, 4, or 8, and the associated probabilities are
1/136, 120/136, and 15/136, so that the expectation for X is EX=4.41.

I want to know what this expectation is for fixed t and variable s as
s increases. I can provide the formula for fixed s and t, but I have
no idea how to investigate the behaviour of this formula as s increases.

[tex]EX=(1+t)\frac{\sum_{i=0}^{s}i\binom{ts}{i}\binom{ts}{ti}}{\sum_{i=0}^{s }\binom{ts}{i}\binom{ts}{ti}}[/tex]


Register to reply

Related Discussions
Casio fx-9860G - calculating binomial coefficients and binomial distribution General Math 3
Binomial Coefficients Set Theory, Logic, Probability, Statistics 2
Binomial coefficients Calculus & Beyond Homework 2
Binomial Coefficients Calculus & Beyond Homework 6
Binomial Coefficients Introductory Physics Homework 4