Question

In a class of 50 students it was found that 27 study History ,27 study civics,25 study economics, 15 study both history and civics , 16 study both civics and economics , 14 study both history and economics , 9 study all the three subjects. Find
1. How many study Economics only
2. How many students do not study any of the three subjects
3. How many study Economics but not civics.

Answer 1

History $=27$ $\left(H\right)$

Civics $=27$ $\left(C\right)$

Economics $=25$ $\left(E\right)$

History & Civics $=15[H\cap C]$

Civics & Economics $=16[C\cap E]$

History & Economics $=14[H\cap E]$

$9^{}$ study all

1) Only Economics $=E-(H\cap E)-(C\cap E)+(H\cap C\cap E)$

$=25-16-4+9=4$ (Answer)

2) Only Economics $=4$

So, Only History $=27-15-14+9=7$

& Only Civics $=27-16-15+9=5$

Total students who studies $=4+7+5+H\cap C+C\cap E+H\cap E-9-9-9$

$=-11+15+16+14$

$=4+16+14$

$=34$

3) Studies Economics but not Civics $=4+14-9=9$ (Answer)