Monochromatic-triangle counting formula

In a red-blue edge colouring of $K_n$, let $d_v$ be the red degree of vertex $v$ and let $M$ be the number of monochromatic triangles. Every nonmonochromatic triangle has exactly two vertices at which its two incident edges have different colors, soSince $d_v(n-1-d_v)\le \lfloor (n-1{)}^2/4\rfloor$, this identity gives a lower bound for $M$.