Red and White Cells 1

Let $A_{m,n}$ be $0$ if the colour of the cell on row $m$ column $n$ is red or otherwise $1$ where $1\le m,n\le150$ ,and $R_m=\sum^{150}_{j=1}A_{m,j}, C_n=\sum^{150}_{i=1}A_{i,n}, S=\sum^{150}_{i=1}R_i=\sum^{150}_{j=1}C_j$ $\begin{aligned}T&=\sum^{150}_{i=1}\sum^{150}_{j=1}(1-A_{i,j})R_iC_j\\&\le\frac{1}{2}\sum^{150}_{i=1}\sum^{150}_{j=1}(1-A_{i,j})(R^2_i+C^2_j)\\&=\frac{1}{2}\bigg(\sum^{150}_{i=1}R^2_i\sum^{150}_{j=1}(1-A_{i,j})+\sum^{150}_{j=1}C^2_j\sum^{150}_{i=1}(1-A_{i,j})\bigg)\\&=\frac{1}{2}\bigg(\sum^{150}_{i=1}R^2_i(150-R_i)+\sum^{150}_{j=1}C^2_j(150-C_j)\bigg)\\&=2\bigg(\sum^{150}_{i=1}\frac{R_i}{2}\cdot\frac{R_i}{2}\cdot(150-R_i)+\sum^{150}_{j=1}\frac{C_j}{2}\cdot\frac{C_j}{2}\cdot(150-C_j)\bigg)\\&\le2\Bigg(\sum^{150}_{i=1}\bigg(\frac{\frac{R_i}{2}+\frac{R_i}{2}+(150-R_i)}{3}\bigg)^3+\sum^{150}_{j=1}\bigg(\frac{\frac{C_j}{2}+\frac{C_j}{2}+(150-C_j)}{3}\bigg)^3\Bigg)\\&=2(150\cdot50^3+150\cdot50^3)\\&=75000000\end{aligned}$ Equality holds when $R_i=C_j=100$ for $1<i,j<150$ .

The following is an example: The cell in row $m$ column $n$ is red if $m-n\equiv1,2,3,...,50\pmod{150}$ or otherwise white.