­At Rocket Brown Elementary School there are 150 students and

­With 96% incorrect out of first 24 timers taken, the question may seem to be very tough, but can be simplified by Venn Diagram.

Otherwise, the solution would be 

Different components: Only h, Only t, Only f, Only h&t, Only h&f, Only t&f, all three, None

Given:
Total = Only h + Only t + Only f + Only h&t + Only h&f + Only t&f+ all three + None  = 150
H = Only h + Only h&t + Only h&f + all three = 40
T = Only t + Only h&t + Only t&f + all three = 50
F = Only f + Only f&t + Only h&f + all three = 90
Only h&t + Only h&f + Only t&f = 70

So, just re arranging the total....
H+T+F=40+50+90 = 180
(Only h + Only h&t + Only h&f + all three) + (Only t + Only h&t + Only t&f + all three) + (Only f + Only f&t + Only h&f + all three) = 180
Only h + Only t + Only f + 2(Only h&t + Only h&f + Only t&f) + 3(all three)  = 180
(Only h + Only t + Only f + Only h&t + Only h&f + Only t&f + all three) + (Only h&t + Only h&f + Only t&f) + 2*all three  = 180
(150 - None) + 70 + 2*all three = 180
 None - 2* all three = 40

As we increase None, 'all three will also increase'. Thus, for All \(three_{max}\), we will get \(None_{max}\)
As 90 play football and 90 is largest set of people given, the none could be all remaining, that is 150-90 = 60.
So 60 - 2*all three = 40 or all three = 10.

Note: If someone wants to know all the parts. 
Since everything is a part of F, all parts that do not contain F are 0.
Total = Only h + Only t + Only f + Only h&t + Only h&f + Only t&f+ all three + None  = 150
H = Only h + Only h&t + Only h&f + all three = 0 + 0 + Only h&f + 10 = 40........Only h&f = 30
T = Only t + Only h&t + Only t&f + all three =  0 + 0 + Only t&f + 10 = 50........Only t&f = 40
F = Only f + Only f&t + Only h&f + all three =  Only f + 30 + 40 + 10 = 90........Only f = 10
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