In a group of 68 students, each student

[jadorexox]

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?

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Hi guys, I actually got the right answer for this problem in my CAT exam, but wanted to confirm if my method of solving it is actually accurate or if I just "got lucky."

History 25
Math 25
English 34

Subtract 3 from each number is 3 people are in all three classes

History 25-3 = 22
Math 25-3 = 22
English 34-3 = 31

Additionally, subtract 3 from the original 68 number since you want to "get rid of them" while you count for the pool.

68-3 = 65

Add up 22+22+31 to get 75

75-65 = 10

Appreciate your help!!

[Sage Pearce-Higgins]

That looks good to me. For 3-set problems the double-set matrix doesn't work (unless you did it in 3-d, which would be confusing). Although the explanation shows a Venn diagram, I prefer to think of the "overlap" between sets in terms of "tickets". Pretend that each student got a ticket for attending a class, then clearly the number of tickets would exceed the number of students if some students took more than one class.

In this case, we have 68 students and 25 + 25 + 34, that's 84 tickets. Now, the 3 really enthusiastic students who are in all classes have 9 tickets between them. That makes 75 tickets left for 65 students. So those extra 10 tickets must belong to students taking 2 classes.

From this, we can derive a formula:
Number of students is equal to total number of "tickets" - 2 x number of students taking 3 classes - number of students taking 2 classes