Paths on a grid

Question

How many paths of minimum length are there from $A$ to $B$ in the grid below?

I thought we should have 16+30+40+30+16 = 132 if we divide the grid into a number of small rectangles but that turned out to be wrong.

Answer 1

For now, lets ignore the two missing squares on the corners.

You will notice that with any path of shortest length, there are always $4$ moves to the Right (R) and $4$ moves Up (U).

For example, one possible path is UURURRUR.

i.e. it is a matter of counting the number of such combinations.

This is given by $C_4^8=70$ since you choose $4$ out of the $8$ positions to place the "U", and the remaining must all be R.

However, two corners are missing, so you have to subtract the number of paths that used the corners which is just $2$.

Hence the final answer is $68$.