Marvin the fly starts at (0,0) Each step, Marvin moves one unit right or one unit up. He is trying to get to the point (6,7). However, at \((4,3)\) there is a frog that will eat him if he goes through that point. In how many ways can Marvin reach (6,7)?
+1223 First, find the total number of ways to get from (0, 0) to (6, 7). There are 6 horizontal steps and 7 vertical steps for a total of 13 steps, so there are \(\binom{13}{6} = 1716\) ways.
Then, find the number of paths that pass through (4, 3). From (0, 0) to (4, 3), there are 4 horizontal steps and 3 vertical steps for a total of 7 steps. \(\binom{7}{4} = 35\)
We then calculate the number of paths from (4, 3) to (6, 7). There are 2 horizontal steps and 4 vertical steps for a total of 6 steps, giving \(\binom{6}{2} = 15\). Overall, there are 35 * 15 = 525 paths that pass through (4, 3), where the frog is. We don't want that, so we subtract this number from the total number of paths to get the answer.
