You are a chief for an electric utility company. The employees in your section cut down trees, climb poles, and splice wire. You report that of the 128 employees in your department 10 cannot do any of the three (management trainees), 25 can cut trees and climb poles only, 31 can cut trees and splice wire, but not climb poles, 18 can do all three, 4 can cut trees only, 3 can splice wire, but cannot cut trees or climb poles, and 9 can do exactly one of the three. How many employees can do at least two of the three jobs mentioned?

[tex]n(P\cup T \cup W)'=10\\n(P \cap T \cap W')=25\\n(P' \cap T \cap W)=31\\n(P \cap T \cap W)=18\\n(P' \cap T \cap W')=4\\n(P' \cap T' \cap W)=3[/tex]

Since 9 can do exactly one of the three.

[tex]n(P' \cap T \cap W')+(P' \cap T' \cap W)+n(P \cap T' \cap W')=9\\4+3+n(P \cap T' \cap W')=9\\n(P \cap T' \cap W')=9-7\\$Number of those who climb pole only, n(P \cap T' \cap W')=2[/tex]

[tex]U=n(P \cap T' \cap W')+n(P' \cap T' \cap W)+n(P' \cap T \cap W')+n(P \cap T \cap W')\\+n(P' \cap T \cap W)+n(P \cap T' \cap W)+n(P \cap T \cap W)+n(P\cup T \cup W)'[/tex]

[tex]128=2+3+4+25+n(P' \cap T \cap W)+31+18+10\\128=93+n(P' \cap T \cap W)\\n(P' \cap T \cap W)=128-93=35[/tex]

Number of Employees who can do at least two of the jobs

[tex]=n(P \cap T \cap W')+n(P' \cap T \cap W)+n(P \cap T' \cap W)+n(P \cap T \cap W)\\=25+35+31+18\\=109[/tex]

109 Employees can do at least two of the jobs.

OR

9 can do exactly one job
10 cannot do any one of the jobs

Therefore: Number who can do at least two jobs

=128-(10+9)=109