Final answer:
To find the number of up/right lattice paths from (0, 0) to (10, 10) via (4, 5) but not through (6, 8), we calculate the paths for each segment using the binomial coefficient and then subtract the number of paths that would pass through (6, 8).
Step-by-step explanation:
The student has asked to determine the number of up/right lattice paths from point (0, 0) to point (10, 10) passing through point (4, 5) but not through point (6, 8). To solve this, we can break the problem into two segments: (0, 0) to (4, 5) and (4, 5) to (10, 10), then we multiply the number of paths for each segment.
Each segment is a simple combinatorics problem that can be solved using the binomial coefficient, which is calculated as C(n, k) = n! / (k! * (n-k)!), where n is the total number of steps and k is the number of steps in one direction.
However, since we must avoid the point (6, 8), we need to subtract the paths that would pass through this point. To find these paths, we would calculate the number of ways to get from (0, 0) to (6, 8) and then from (6, 8) to (10, 10) and subtract this from our total.