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The total number of students is given by
\(a + b + c + d + e + f + g + h = 75.\)
Of these,
\(a + d + f + g = 17\) are enrolled in Physics,
\(b + d + e + g = 28\) are enrolled in Chemistry, and
\(c + e + f + g = 39\) are enrolled in Biology.
Adding these three equations gives \(( a + b + c ) + 2( d + e + f ) + 3g = 84.\)
We also know that
\(d + g = 5\) study both Physics and Chemistry,
\(e + g = 7\) study both Chemistry and Biology, and
\(f + g = 6\) study both Physics and Biology.
Adding these three questions yields \(( d + e + f ) + 3g = 18.\)
Since \(4\) students are enrolled in all three classes, we have \(g = 4,\)
Plugging this value for \(g\) into the above equation yields
\((d + e + f) + 3g = 18\)
\((d + e + f) + 12 = 18\)
\(d + e + f = 6.\)
So,
\(( a + b + c ) + 2( d + e + f ) + 3g = 84\) yields
\((a + b + c) + 2(6) + 3(4) = 84\)
\((a + b + c) + 12 + 12 = 84\)
\(a + b + c = 60.\)
Finally, using the equation, \(a + b + c + d + e + f + g + h = 75\), we see that
\((a + b + c) + (d + e + f) + g + h = 75\)
\(60 + 6 + 4 + h = 75\)
\(h = 5.\)
Therefore, the answer is D.
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