How many backpacks out of 38 contained none of the three writing instruments?

First, I would like to say that the whole question is not here. The question states: 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?

OK I will explain this the best I can!

Starting out, their are 10 backpacks that contain all the items. 12 contained both a black and blue pen. You would subtract 10 from 12; therefore there are 2 backpacks that contain only black and blue pens. 18 contain both a blue pen and a pencil. subtract 10 from 18 and you get 8. 15 contained both a black pen and a blue pen. You would subtract 10 from 15 and you get 5. There are 23 with black pens. So you would do 23 - (5+10+2)= 6. There are 6 backpacks that contain only black pens. There are 27 that contain blue pens. So you would do 27 - (5+10+8) = 4. There are 4 backpacks that contain only blue pens. There are 21 backpacks that contain a pencil. You would do 21 - (2+10+8) = 1. All together, there a 6 that contain only black pens, 5 that contain only black and blue pens, 4 that contain only blue pens, 10 that contain all three items, 2 that contain only black pens and pencils, 8 that contains only blue pens and pencils, and 1 that contains only 1 pencil. So you would add 6+5+4+10+8+2+1= 36. There are 38 backpacks, so you would subtract 36 from 38. There are 2 backpacks that contain none of the three writing instruments.

Ashfords quiz says this is right.