Fran the fly starts at (0,0) and each minute moves one unit right or one unit up. She is trying to get to the point (5,7). However, at (4,3) there is a spider that will eat her if she goes through that point. In how many ways can Fran reach (5,7)?
Well, I don't really know this but I'll give it a shot:
We need to caculate the amount of ways from (0,0) to (5,7): 12C5=792 ways from (0,0) to (5,7), and then subtract that by the amount of ways that pass the spider.
Next, we also need to caculate the paths from (0,0) to (4,3) and multiply that by the amount of paths from (4,3) to (5,7):
7C3 * 5C1 = 175.
792 - 175 = 617.