There is a robot on a 0-indexed $m \times n$ grid. The robot is initially located at the top-left corner (i.e., origin, grid[0][0]). The robot tries to move to the bottom-right corner (i.e., destination, grid[m - 1][n - 1]). The robot can only move either down or right at any point in time. In addition, there is a set of k points $C=\{grid[x_1][y_1],\cdots,grid[x_k][y_k]\}$. The question is what is the expected number of points on a path from grid[0][0] to grid[m - 1][n - 1].
If we are talking about the total number of paths, it is a classical dynamic programming question Unique Paths. To compute the expected number of points on a path, we can enumerate every path and count the points on each path. This algorithm will be very slow, since there will be many paths.
The reviewer's hint seems to indicate there is some closed-form solution for the answer, which I failed to find. The hint is roughly like for every point, count how many paths from the origin to this point, and also count how many paths from this point to the destination. But I am not sure how to use this to get a closed-form solution on the expected number of points on a path from origin to destination.
You can rewrite the desired random variable in the form $X = I_1 + \dotsc + I_k$, where $I_i$ stands for the indicator of the event that a random path passes through the $i$-th of the points. It remains to use the linearity and to count the probability of the event "a random path passes through $i$-th point", which is some easy combinatorics.