16. (25 points) A competition consists of $n$ true/false questions. After tallying the answers of eight candidates, it was found that: for any two questions, exactly two candidates answered “T, T”; exactly two candidates answered “F, F”; exactly two candidates answered “T, F”; and exactly two candidates answered “F, T”. Find the maximum value of $n$.


16. Let “ $T$ ” be 1, and “ $F$ ” be 0. Thus, we get an 8-row $n$-column number table.

Obviously, swapping 0 and 1 in the same column does not change the nature of the table. Therefore, we can assume that the first row is all 0s.
Let the $i$-th row have $a_{i}$ zeros $(i=1,2, \cdots, 8)$.
Then $a_{1}=n, \sum_{i=2}^{8} a_{i}=4 n-n=3 n$.
Next, consider the number of "00" pairs in the same row, then
$$
\begin{array}{l}
\sum_{i=1}^{8} \mathrm{C}_{a_{i}}^{2}=2 \mathrm{C}_{n}^{2} \Rightarrow \sum_{i=2}^{8} \mathrm{C}_{a_{i}}^{2}=\mathrm{C}_{n}^{2} \\
\Rightarrow \sum_{i=2}^{8} a_{i}^{2}-\sum_{i=2}^{8} a_{i}=n^{2}-n .
\end{array}
$$

By the Cauchy-Schwarz inequality,
$$
\sum_{i=2}^{8} a_{i}^{2} \geqslant \frac{1}{7}\left(\sum_{i=2}^{8} a_{i}\right)^{2},
$$

we know $\frac{1}{7}\left(\sum_{i=2}^{8} a_{i}\right)^{2}-\sum_{i=2}^{8} a_{i} \leqslant n^{2}-n$
$\Rightarrow \frac{1}{7}(3 n)^{2}-3 n \leqslant n^{2}-n$
$\Rightarrow n \leqslant 7$.
Table 1 is the case when $n$ takes its maximum value.

Thus, the maximum value of $n$ is 7.