no. of rectangles is pC2 * qC2 for obvious reasons

 

lets consider a right as the letter R and up as the letter U

Now for a person to move from one diagonal to the other, he has to move thro p-1 rights and q-1 ups i.e. its eqvt to arranging a sequence of p-1 R's and q-1 U's

So no. of ways is ((p-1)+(q-1))!/(p-1)!(q-1)!