Since this was asked under "Pascal's Triangle and Binomial Expansion" let's use that
Term Multiplier/Pascal's Triangle Expansion Based on Exponent
(a+b)^0: 1
(a+b)^1: 1 - 1
(a+b)^2:1 - 2 - 1
(a+b)^3: 1 - 3 - 3 - 1
(a+b)^4:1 - 4 - 6 - 4 - 1
(a+b)^5:1 - 5 - 10 - 10 - 5 - 1
(a+b)^color(red)(6): 1 - 6 - 15 - color(red)(20) - 15 - 6 - 1
color(white)("XXXX")That is, the expansion of
color(white)("XXXX")color(white)("XXXX")(a+b)^(color(red)(6))
color(white)("XXXX")is
color(white)("XXXX")color(white)("XXXX")1a^6b^0+6a^5b^1+15a^4b^2+color(red)(20a^3b^3)+15a^2b^4+6b^1b^5+1a^0b^6
with a= x/2 and b=-2y
the middle term of (x/2 - 2y)^6 is
color(red)(20)(x/2)^3(-2y)^3
= -20x^3y^3
