3. A math competition has a total of 15 questions. The table below shows the number of people who got $n$ $(n=0,1,2, \cdots, 15)$ questions correct:
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
$n$ & 0 & 1 & 2 & 3 & $\cdots$ & 12 & 13 & 14 & 15 \\
\hline Number of people who got $n$ questions correct & 7 & 8 & 10 & 21 & $\cdots$ & 15 & 6 & 3 & 1
\end{tabular}

If it is also known that the average number of questions answered correctly by students who answered 4 or more questions correctly is 6, and the average number of questions answered correctly by students who answered 10 or fewer questions correctly is 4. How many people does this table at least include?
(Tip: The number of people who got at most 3 questions correct is $7+8+10+21$ $=46$ people.
The number of people who got at least 12 questions correct is $15+6+3+1=25$ people.
The total number of questions correctly answered by those who got at most 3 questions correct is 91 questions,
The total number of questions correctly answered by those who got at least 12 questions correct is 315 questions.
Let the total number of people be $y$, and the number of people who got 11 questions correct be $x_{11}$, then the number of people who got at least 4 questions correct is $y-46$, and the number of people who got at most 10 questions correct is $y-25-x_{11}$. Therefore, the total number of questions correctly answered by those who got 5 to 11 questions correct is
$$
\begin{array}{l}
6(y-46)-315=4\left(y-x_{11}-25\right)+11 x_{11}-91 \\
y=200+3.5 x_{11}\left(x_{11} \geqslant 0\right) . \\
\text { When } \left.x_{11}=0 \text {, } y=200 \text { is the minimum. }\right)
\end{array}
$$