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?
3
11
2
15
I came up with 2. IS that correct?
My Venn diagram has all my circles totaling 35 , so that 3 contained none of the above
My circles contained the following entries:
Black pen: 6 5 10 2
Blue pen : 3 8 10 5
pencil : 1 2 10 8
See if that matches with yours
To solve this problem, we can use the principle of inclusion-exclusion.
Let's start by counting the number of backpacks that contain at least one writing instrument. We can add up the number of backpacks that contain a black pen, a blue pen, or a pencil and then subtract the overlaps.
Number of backpacks with at least one writing instrument = Number with a black pen + Number with a blue pen + Number with a pencil - Number with both a black pen and a blue pen - Number with both a black pen and a pencil - Number with both a blue pen and a pencil + Number with all three items.
Plugging in the given values:
Number of backpacks with at least one writing instrument = 23 + 27 + 21 - 15 - 12 - 18 + 10 = 56.
So, out of the 38 backpacks, 56 contained at least one writing instrument.
To find the number of backpacks that contained none of the three writing instruments, we can subtract this result from the total number of backpacks:
Number of backpacks with none of the three writing instruments = Total number of backpacks - Number of backpacks with at least one writing instrument = 38 - 56 = -18.
Since we can't have a negative number of backpacks, the number -18 doesn't make sense in the context of this problem. It's possible there may be an error in the given information or the problem itself. Please double-check the conditions and try again.