Non-leptonic kaon decays at large $N_c$

We study the scaling with the number of colours, $N_c$, of the weak amplitudes mediating kaon mixing and decay. We evaluate the amplitudes of the two relevant current-current operators on the lattice for $N_c=3-7$. We conclude that the subleading $1/N_c$ corrections in $\hat{B}_K$ are small, but those in the $K \rightarrow \pi\pi$ amplitudes are large and fully anti-correlated in the $I=0, 2$ isospin channels. We briefly comment on the implications for the $\Delta I=1/2$ rule.