2012 IMO Shortlist C3

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\left(C_{1}, C_{2}, C_{3}\right)$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain.

在 $999 \times 999$ 的方桌中，一些单元格为白色，其余单元格为红色。令 $T$ 为单元格 $\left(C_{1}, C_{2}, C_{3}\right)$ 的三元组数，前两个在同一行，后两个在同一列，其中 $C_{1}$ 和 $C_{3}$ 为白色，$C_{2}$ 为红色。找出 $T$ 可以达到的最大值。