One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and applying $\mathrm{Hom}(-,N)$.
I'm struggling to remember which is which and I'm sure there is a(n obvious?) reason. Why does it have to be projective + $\mathrm{Hom}(-,N)$ or injective + $\mathrm{Hom}(M,-)$ but cannot be projective + $\mathrm{Hom}(M,-)$?