Question 10: In a training session, 17 participants have laptops, 11 have tablets, 9 have laptops and tablets, 6 have laptops and books, and 4 have both tablets and books. Eight participants have all three items. The total number of participants with laptops, tablets, or books is 35. How many participants have books?
Solution:
Let, \(L=\text{Participants having laptops}\; ;\)\(T=\text{Participants having tablets}\; ;\)\(\;B=\text{Participants having Books}\)
\(n(L \cup T \cup B)=35\)
\(n(L)=17\)
\(n(T)=11\)
\(n(B)=?\)
\(n(L\cap T)=9\)
\(n(L\cap B)=6\)
\(n(B \cap T)=4\)
\(n(L\cap T \cap B)=8\)
We know that
\(n(L\cup T\cup B)=\ \)\(n(L)+n(T)+n(B)\ \)\(−\ n(L\cap T)−n(L\cap B)\ \)\(−\ \)\(n(B\cap T)+n(L\cap T\cap B)\)
\(35=\ \)\(17+11+n(B)\ \)\(−\ 9−6\ \)\(−\ \)\(4+8)\)
\(35=\ \)\(17+n(B)\)
\(n(B)=35-17\)
\(n(B)=18\)
\(\boxed{18 \text{ participants have books}}\)