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?

Answer 1

Answer:2

Step-by-step explanation:

Given

Total bags=38

No of bags containing black pen is
`n\left ( A\right )23`

No of bags containing blue pen
`n\left ( B\right )27`

No of bags containing pencil
`n\left ( C\right )21`

no of bags containing both Black pen & blue
`n\left ( A\cap B\right ) 15`

no of bags containing both Black pen & Pencil
`n\left ( B\cap C\right ) 12`

no of bags containing both Blue pen & Pencil
`n\left ( A\cap C\right ) 18`

no of bags containing all
`n\left ( A\cap B\cap C\right ) is  10`

using Venn diagram

`n\left ( A\ \cup B\ \cup C\right )=n\left ( A\right )+n\left ( B\right )+n\left ( C\right )-n\left ( A\cap B\right )-n\left ( B\cap C\right )-n\left ( A\cap C\right )+n\left ( A\cap B\cap C\right )`

`n\left ( A\ \cup B\ \cup C\right )=23+27+21-15-12-18+10`

`n\left ( A\ \cup B\ \cup C\right )=36`

and we know
`n\left ( A\ \cup B\ \cup C\right )+n\left ( empty\ bag\right )=38`

`n\left ( empty\ bag\right )=38-36=2`

Therefore there are 2 bags which are empty.