Chess 40
Drama 30
Math 25

The table above shows the number of students in 3 clubs at McAuliffe School. Although no student is in all 3 clubs, 0 students are in both Chess and Drama, 5 are in both chess and math, and 6 are in both drama and math. How many different students are in the 3 clubs.

A 68
B 69
C 74
D 79
E 84

Solution: Answer C

The solution show a Venn Diagram and adds up ALL the parts - i.e. 25+14+14+5+10+6. But shouldn't the solution be the 25+14+14, since these are the ones that show all the DIFFERENT students?

Thank you!

Is there a mistake in the question? Are there 10 students in both Chess and Drama?

According to the rest of the info given, there must be a typo - the poster must have meant to say there were 10 students in both chess and drama.

If you add only 25, 14, and 14, you will not be adding all of the different students. Take a look at the Venn diagram again before you read the rest of this. The 25 in the big portion of the Chess circle represents the students who are taking ONLY chess. It does not represent any of the students who are taking chess plus something else. Ditto with each of the two 14's. We don't want the total number of students who are in ONLY ONE club - we want to count all of them, including the ones in 2 clubs.

So 25+14+14 is the starting point, but now we have to add the students who are in 2 clubs. There are 10 students in both chess and drama, so add 10. 6 students are in both drama and math, so add 6. 5 students are in both chess and math, so add 5.

Notice that the purpose of the Venn diagram is to split out all of the categories separately - those who are in exactly one club, those who are in exactly two clubs, and those who are in all three clubs (zero, in this case).