A combinatorics problem

Finding for the even number of rectangles is also the same as finding the number of rectangles minus the odd number of rectangles

there are 8 vertical lines and 7 horizontal lines or vice versa, choosing 2 from 8 vertical lines and 2 from 7 horizontal lines will give us a rectangle

finding for the number of rectangles in a 7 by 6 rectangle= $(8C2)(7C2)$

$\Rightarrow 28\cdot 21= 588$

finding for odd number of rectangles in a 7 by 6 rectangle= $(7+5+3+1)(6+4+2)$

$\Rightarrow 16\cdot 12=192$

finding for even number rectangles = $588-192$

$\Rightarrow \boxed{396}$

Or either we can do this by giving coordinate no with origin at vertex and finding no of rectangles with even length + no of rectangles with even breath - no of rectangles with even length and even breath = 252+252-108=396