Full Solution to the Three -Question Quiz Problem

First a re-statement of the brainstretcher:

Thirty students took a three-question quiz with the following results:

20 got the first question correct
16 the second question
10 the third question
11 got the first and second correct
7 the first and third
5 the second and third
4 got all three correct

How many missed all three?

For this we will eschew set theory and tempting as it is a Venn diagram (plus I don't know how to draw a Venn diagram in my blog).  We will just be very good little accountants.

Start with the original 30 students and subtract out those that got each of the problems correct.

30 - (20 + 16 + 10), of course this is negative but hold your horses.

Now add back in the ones who solved two problems because they were counted twice in the first number that was subtracted.

30 - (20 + 16 + 10) + (11 + 7 + 5)

But wait, you say!!!!  What about the ones who got all three questions correct?

AHA!!! this number was accounted for all over the place in our last expression: three times in the first parenthesis (subtracted), and three times in the second parenthesis (added).  They cancel each other out.  So we only have to subtract out the last 4 to make our expression complete.

30 - (20 + 16 + 10) + (11 + 7 + 5) - 4 = 3

Therefore only 3 students failed to solve any of the problems