jmc

We know that we must take a 7 unit path. If we look at the grid a little more carefully, we can see that our path must consist of 4 steps to the right and 3 steps up, and we can take those steps in any order. So in order to specify a path, we must choose 3 of our 7 steps to be `up' (and the other 4 steps will thus be `right'). Hence the number of paths is $$\left(\genfrac{}{}{0ex}{}{7}{3}\right)=\frac{7\times 6\times 5}{3\times 2\times 1}=\boxed{35}.$$