= Solution
The <giant magnetoresistance> mechanism is spin-dependent conduction through the two ferromagnets. Exchange splitting gives majority and minority $d$ bands different <density of states> values at the <Fermi energy>. Scattering into those bands, together with interface scattering, produces different resistances for the two spin channels. The relevant quantity is the spin-dependent transport/scattering rate, not simply a count of all available states.
For parallel layer <magnetizations>, one global spin channel encounters relatively weak scattering in both Co and NiFe, providing a low-resistance path. For antiparallel <magnetizations>, each global spin channel encounters one more strongly scattering ferromagnet, so both paths are comparatively resistive. The channels conduct in parallel while each crosses the layers in series. This is <current-perpendicular-to-plane magnetoresistance>, because the current crosses the interfaces.<Nobel Lecture: Origin, development, and future of spintronics by Albert Fert>[Spin-dependent conduction and the distinction between CPP and CIP transport]{ref}
In the simplest <two-current resistor model>, with equal magnetic layers having channel resistances $a,b$ and spacer resistance $r$ per channel,
$$
R_P=\frac{(2a+r)(2b+r)}{2(a+b+r)},\qquad
R_{AP}=\frac{a+b+r}{2},\qquad
R_{AP}-R_P=\frac{(a-b)^2}{2(a+b+r)}.
$$
Real Co/NiFe layers need not have equal parameters, but the same-sign spin asymmetries normally give $R_{AP}>R_P$. Their different <magnetic coercivities> allow the two configurations to be selected separately.
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