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 $$a+b$$.

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