Red and White Cells 2

Let $A_{n}$ be $0$ if the colour of the $n^{th}$ cell from the left is red or otherwise $1$ where $1\le m,n\le150$ and $S=\sum^{150}_{i=1}A_i$

Lemma: $(1-A_i)k=(1-A_i)(k+A_i)$ Proof: $\begin{aligned}R.H.S.-L.H.S.&=(1-A_i)(k+A_i)-(1-A_i)k\\&=(1-A_i)A_i\end{aligned}$ As $A_i=0 or 1$ , $R.H.S.-L.H.S.=0$ so $L.H.S.=R.H.S.$ .

Then $\begin{aligned}T&=\sum^{150}_{i=1}\bigg((1-A_i)\sum^{i-1}_{j=1}A_j\sum^{150}_{j=i+1}A_j\bigg)\\&\le\sum^{150}_{i=1}\Bigg(\Big(1-A_i)(\frac{\sum^{i-1}_{j=1}A_j+\sum^{150}_{j=i+1}A_j}2\Big)^2\Bigg)\\&=\sum^{150}_{i=1}(1-A_i)\bigg(\frac{S}{2}\bigg)^2\\&=(150-S)\bigg(\frac{S}{2}\bigg)\bigg(\frac{S}{2}\bigg)\\&\le\Bigg(\frac{(150-S)+\frac{S}{2}+\frac{S}{2}}{3}\Bigg)^3\\&=50^3\\&=125000\end{aligned}$

Here is an example of how could we colour to attain $T=125000$ . We could colour the $50$ cells from left and $50$ cells from right white, the remaining $50$ red.