Quicker erh... sort of.. but it's a bit ad hoc:
(y+1/y)^10 (y-1/y)^7 = (y^2 - 1/y^2)^7 (y+1/y)^3
the first bracket has powers (14,10,6,2,-2,-6,-10,-14) and the 2nd one has
(3,1,-1,-3) so the only ways to get a power of -3 is to match -2 with -1 and 3 with -6
so it becomes (3 C 2) * (7 C 4) +...