  In 2014, T. Komatsu and L. Szalay studied the balancing binomial
coefficients. In this paper, we focus on the following Diophantine equation
$$\binom{1}{5}+\binom{2}{5}+...+\binom{x-1}{5}=\binom{x+1}{5}+...+\binom{y}{5}$$
where $y>x>5$ are integer unknowns. We prove that the only integral solution is
$(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic
logarithms.
