a$.$ How many paths are there that move from point $A$ to point $B$ along the lines shown in the grid below, using only

movements that go down or to the right?

b$.$ How many paths are there that move from point $A$ to point $B$ along the lines shown in the grid below, using only

movements that go down or to the right, which pass through the green box? $($Hint: think about paths from A to

the green, and paths from the green to B$.)$