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overlapping sets

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by rommysingh » Thu Aug 06, 2015 6:51 pm
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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by GMATGuruNY » Thu Aug 06, 2015 7:00 pm
I posted a solution here:

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by [email protected] » Thu Aug 06, 2015 11:08 pm
Hi rommysingh,

3-Group Overlapping Sets questions are relatively rare on the Official GMAT (you likely will NOT see this version of Overlapping Sets on Test Day). However, there is a formula that you can use to solve it.

Total = (1st group) + (2nd group) + (3rd group) - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd) - 2(all 3 groups).

In overlapping sets questions, any person who appears in more than one group has been counted more than once. When dealing with groups of people, you're not supposed to count any individual more than once, so the formula 'subtracts' all of the extra times that a person is counted.

For example, someone who is in BOTH the 1st group and the 2nd group will be counted twice....that's why we SUBTRACT that person later on [in the (1st and 2nd) group].

In this prompt, we're given the Total, a number for each of the 3 individual groups and the number of people who appear in all 3 groups. The equation would look like this...

68 = 25 + 25 + 34 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)- 2(3)

68 = 84 - 6 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

68 = 78 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

(1st and 2nd) + (1st and 3rd) + (2nd and 3rd) = 10

Since the prompt asks for the total number of students that are in exactly 2 classes, we have our answer.

Final Answer: [spoiler]B; 10[/spoiler]

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