, 2019

Solution:

For a lattice point $(x, y)$, let $F(x, y)$ denote the number of up-right paths from $(0,0)$ to $(5,5)$ that don't pass through $(x, y)$, and let
$$S=\sum_{0 \leq x \leq 5} \sum_{0 \leq y \leq 5} F(x, y)$$
Our answer is $\frac{S}{36}$, as there are 36 lattice points $(x, y)$ with $0 \leq x, y \leq 5$.
Notice that the number of up-right paths from $(0,0)$ to $(5,5)$ is $\binom{10}{5}=252$ because each path consists of 10 steps, of which we can choose 5 to be to the right. Each of these paths passes through 11 lattice points $(x, y)$ with $0 \leq x, y \leq 5$, so each path contributes $36-11=25$ to the quantity we are counting in $S$. Then $S=25 \cdot 252$, so our answer is $\frac{25 \cdot 252}{36}=175$.