Eve randomly chooses two **distinct** points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\binom{20}{10} \cdot \frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.

Let's think step by step and output the final answer within \boxed{}.
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