Problem Solving

In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?

a)13
b)10
c) 9
d) 8
e) 7

[spoiler]Ans : 10 [/spoiler]

My approach :

We need to find HM+HE+ME

68 = H+M+E-(HM+HE+ME) + HME
68 = 25+25+34 - (HM+HE+ME) + 3

68 = 84 - (HM+HE+ME) + 3

HM+HE+ME = 19

What am i missing ??

We know that 3 students are in all 3 courses so that means we are only looking at 65 students. Subtracting these students leads to 22 History, 22 Math, and 31 English. The sum of these groups is 75, thus 10 people is the overlap.

In your approach you are double counting people by adding 3 additional students to the ones that are registered.

If you simply draw a venn diagram, things become very simple..

Look at the figure below..


Now just write the equation as you see it in the diagram..

i.e.

(25 -x - z - 3) + (25 -x -y -3) + (34 -y -z -3) + x + y + z + 3 = 68

=> x + y + z = 10

thanx DD n Ross , venn diagram makes it clear