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

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.