Let $r$ be the number of rows, and $c$ be the number of columns. Then there are $14r$ boys, and $10c$ girls. So we have the total number of seats $rc=14r+10c+3$ or $rc-14r-10c-3=0$. Completing the multiple, we get $(r-10)(c-14)=143=11*13$. So $r-10=11$ and $c-14=13$ or $r=21$ and $c=27$ giving 567 chairs. There is another solution with $r-10=13$ and $c-14=11$ giving 575 chairs.
An example seating 'b' is boy, 'g' is girl, and 'e' is empty.
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