Combination Square

How many path (combination) possibilities are there?
i've a 8 by 8 grid and am to travel from (0,0) to (8,8). However the square area from (2,3) to (4,5) is a garden. i cannot travel into it, but along the perimeter.
How many different ways are there traveling from (0,0) to (8,8) if i can only move right / upwards.
there is only 1 intersection that you cannot travel, a (3,4) inside the garden.
answer
= (8+8)C8 - [(3+4)C3]*[(16-3-4)C(8-3)]
= 16C8 - [7C3]*[9C5]
= 12870 - 35*126
= 8460
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