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 $a+b$ .