Binomial Coefficients and Fractions

Algebra Level 5

\[ 1 + \frac{1}{2}{5\choose3}+\frac{3}{4}{6\choose3}+{7\choose3}+\frac{5}{4}{8\choose3}+\frac{3}{2}{9\choose3}+\frac{7}{4}{10\choose3} = {a\choose b}\]

It is given that \(2b \le a\) and \(a\) is as small an integer as possible. Determine the value of \(a+b\).