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

Quizplus · Accepted Answer

2To find the number of backpacks that contained none of the three writing instruments, we use the principle of inclusion-exclusion. First, we add the total number of backpacks that contained each item: 23 (black pen) + 27 (blue pen) + 21 (pencil) = 71. Then, we subtract the backpacks that were counted twice because they contained two items: 71 - 15 (black and blue pen) - 12 (black pen and pencil) - 18 (blue pen and pencil) = 26. However, the 10 backpacks that contained all three items were subtracted three times, so we need to add them back once to correct this: 26 + 10 = 36. Since there were 38 backpacks in total, the number of backpacks that contained none of the three writing instruments is 38 - 36 = 2.

Upon Examining the Contents of 38 Backpacks, It Was Found