USAMO 1987 Problem 5

$\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}$ is a sequence of $0 \mathrm{~s}$ and $1 \mathrm{~s}$. $\mathrm{T}$ is the number of triples $\left(\mathrm{a}_{i}, \mathrm{a}_{j}, \mathrm{a}_{\mathrm{k}}\right)$ with $\mathrm{i}<\mathrm{j}<\mathrm{k}$ which are not equal to $(0,1,0)$ or $(1,0,1)$. For $1 \leq i \leq n, f(i)$ is the number of $j<i$ with $a_{j}=a_{i}$ plus the number of $j>i$ with $a_{j} \neq a_{i}$. Show that $\mathrm{T}=\mathrm{f}(1)(\mathrm{f}(1)-1) / 2+\mathrm{f}(2)(\mathrm{f}(2)-1) / 2+\ldots+\mathrm{f}(\mathrm{n})(\mathrm{f}(\mathrm{n})-1) / 2$. If $\mathrm{n}$ is odd, what is the smallest value of T?