Upon examining the contents of 38 backpacks, it was found that 23 contained a black pen, 27 contained a blue pen, and 21 contained a pencil, 15 contained both a black pen and a blue pen, 12 contained both a black pen and a pencil, 18 contained both a blue pen and a pencil, and 10 contained all three items. How many backpacks contained none of the three writing instruments?

Question

Upon examining the contents of 38 backpacks, it was found that 23 contained a black pen, 27 contained a blue pen, and 21 contained a pencil, 15 contained both a black pen and a blue pen, 12 contained both a black pen and a pencil, 18 contained both a blue pen and a pencil, and 10 contained all three items.  How many backpacks contained none of the three writing instruments?

anonymous · Answer

http://answers.yahoo.com/question/index?qid=20090717160525AAywK3i might be a starter.

anonymous · Answer

So, we have sets $P_{\text{black}}, P_{\text{blue}}, P_{\text{pencil}}, T$, where $T$ is our universe (all backpacks). We wish to find: $|(P_{\text{black}} \cup P_{\text{blue}} \cup P_{\text{pencil}})^c|$, so we use the principle of Inclusion Exclusion to find $|(P_{\text{black}} \cup P_{\text{blue}} \cup P_{\text{pencil}})|$. It would be:
$$
|A\cup B \cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B \cap C|+|A \cap B\cap C|
$$Which computes to:
$$
23+27+21-15-12-18+10=36
$$So, the cardinality of the complement of this set in our world $T$ is $38-36=2$ which is our final answer.