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 exactly one of the three writing instruments?
A)3
B)11
C)2
D)15

Question

Quizplus · Accepted Answer

Using the principle of inclusion-exclusion, we can find that there are 38 - 10 = 28 backpacks that contain at least one of the items. We can then find the number of backpacks that contain exactly one item by subtracting the number of backpacks that contain two or three items from this total.

Number of backpacks that contain two or three items = 15 (from the given information) + (23 - 15) + (27 - 15) + (21 - 10 - 15 - 12 - 18) = 3

Number of backpacks that contain exactly one item = 28 - 3 = 25

So the best choice is B with the explanation above. To find the number of backpacks containing exactly one writing instrument, we need to subtract the number of backpacks containing two or three writing instruments from the total number of backpacks (38).

Backpacks containing both a black pen and a blue pen: 15 
Backpacks containing both a black pen and a pencil: 12 
Backpacks containing both a blue pen and a pencil: 18

To find the total number of backpacks containing two or three writing instruments, we add these three numbers and subtract the backpacks containing all three items (counted twice in the previous step):

15 + 12 + 18 - 10 = 35

Therefore, the number of backpacks containing exactly one writing instrument is:

38 - 35 = 3

So, the best choice is option B, 11 backpacks.

Upon Examining the Contents of 38 Backpacks, It Was Found