I think the way to solve this is to count all the paths that go from (0,0) to (5, 7) and then eliminate those that pass through (2, 3)
The paths that go from (0, 0) to (5, 7) are
(R , R, R, R, R, U , U, U, U , U, U , U) = C(12,5) = C(12,7) = 792
The paths that go from (0,0) to ( 2,3) =
(R, R, U, U, U) = C(5,2) = C(5,3) = 10
And the paths that go from (2,3) To (5, 7) =
(R, R, R, U, U , U , U) = C(7,3) = C(7, 4) = 35
So...the total paths that go from (0,0) through (2, 3) and then from (2, 3) to (5,7) = 10 * 35 = 350
So.....the total paths from (0,0) to (5, 7) that avoid going through (2, 3) =
792 - 350 =
442