# Combinatorics with Binomials

I've been trying to find analytical solutions for the zeros of the following expression Zeros in terms of variables n, m and f (here I represented two forms of the expression R, I've worked with both but I couldn't find a general solution, the code is intended to be used on Mathematica):

R[x_, n_, m_] :=
x^(m - 1)*(Sum[
Binomial[n - 1, k]*x^(k - m + 1)*(1 - x)^(n - 1 - k), {k, m,
n - 1}] +
n*(Sum[Binomial[n - 1, k]*x^(k - m + 1)*(1 - x)^(n - 1 - k), {k,
0, m - 1}]) + m*Binomial[n - 1, m - 1]*(1 - x)^(n - m))=
1 - (1 - N)* (Sum[
Binomial[n - 1, k]*x^(k)*(1 - x)^(n - 1 - k), {k, 0,
m - 1}]) + m*Binomial[n - 1, m - 1]*x^(m - 1)*(1 - x)^(n - m);

Zeros[x_, n_, m_, f_] := R[x, n, m] - n/f;


I also tried solving the equation by using Reduce[R[x, n, m] - n/F == 0, x] but he couldn't evaluate it and get to a conclusion. I'd really appreciate some suggestions. Thanks for your time.

• Your two expressions for $R$ don't give the same answer. For example, R[x,5,2] gives $-20 x^4+56 x^3-48 x^2+8 x+5$ using the first definition and $-8 x^4+12 x^3+8 x^2-16 x+\frac{4}{x}+1$ using the second. – Misha Lavrov Apr 12 '17 at 4:02
• Sorry i wrote the second expression wrong. I've corrected it now. Thanks for alerting me. – Matt Apr 13 '17 at 0:19