Problem number 11, Matthew the mouse wants to get to a piece of cheese. He can only move horizontally or vertically between any two cells in the direction shown by the arrows. How many different routes can Matthew take to reach a piece of cheese? So to solve this problem, let's look at each of the cells on the diagram, which is shown on the right, and figure out how many ways Matthew can get to each of the cells in the diagram. For example, Matthew is already at the top left cell, so there's only one way to get there. Now let's look at the cell to its right. Matthew can only move down or to the right as looking at the arrows. So the only way Matthew can reach the cells to his right is by going directly right. Note that Matthew cannot go down, right, up, because you're not allowed to go back up after you've already gone down. So the only way to reach the cell to the right of the original cell is to go immediately to the right. Similarly, the cell that's immediately below Matthew, he can only reach by going down. Now let's look at the cell in the center of the figure that's next to the two 1s. That cell, we have to figure out how many ways we can get to it. Well, you can either get to that cell from above or from the right, because those are the only two moves you're allowed to make. So you could either go there from right, down, or from down, right. That is, to figure out the number of ways to get to the cell, you can add the number of ways to get to the cell above it and to the left, because the number of ways to get to the cell above it, all of those ways you can go to that cell and then go down. The number of ways to get to the cell on its left, you can go all of those ways and then go right. This cell should be a 2, because there are two ways to get to this cell, either go from right, down, or down, right. And by the same logic as before, if we look at the cells to the right of this 2, there's only one way to get to them, which is directly going from the right to the original cell label 2. If there are two ways to get to the cell label 2 originally, there are two ways to get to all of these cells in the row, since you have to go right. By the same logic, there are only two ways to get to each of those cells in this column, because you have to go down from the original 2. Finally, let's look at the last unfilled cell. By the same logic as we had before, you can either get to this cell from above or from the left. From above, you can get to that cell in two ways and then go down. From the left, you can get to that cell in two ways and then go right. The number of ways is 2 plus 2, or 4. That means that you can either get to the top piece of cheese in two ways, the middle piece of cheese in four ways, and the bottom piece of cheese in two ways. The total number of ways to get to any piece of cheese is 2 plus 4 plus 2, which is 8. The answer is C, 8.

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