Question

Fifty college teachers are surveyed as to their possession of colour TV, VCR and tape recorder. Of them, $22$ own colour TV, $15$ own VCR and $14$ own tape recorders. Nine of these college teachers own exactly two items out of colour TV, VCR and tape recorders; and one college teachers owns all three. Then how many of the $50$ college teachers own none of three, colour TV, VCR or tape recorder?

• A. $4$
• B. $9$
• C. $10$
• D. $11$

Answer 1

The correct option is C $10$

Let $n\left(A\right)=$ number of teachers who own colour TV.
$n\left(B\right)=$ number of teachers who own VCR.
$n\left(C\right)=$ number of teachers who own own Tape recorders.
$n\left(E_2\right)=$ number of teachers who own exactly two items.
$n\left(E_0\right)=$ number of teachers who own none of three.

Given, $n\left(A\right)=22;n\left(B\right)=15;n\left(C\right)=14;n\left(E_2\right)=9;n(A\cap B\cap C)=1$
Now, $n\left(E_0\right)=50-n(A\cup B\cup C)$
$n(A\cup B\cup C)=\sum n\left(A\right)-\sum n(A\cap B)+\sum n(A\cap B\cap C)$
$=22+15+14-\sum n(A\cap B)+1$
$=52-\sum n(A\cap B)$

$n\left(E_2\right)=\sum n(A\cap B)-3n(A\cap B\cap C)$
$\Rightarrow 9=\sum n(A\cap B)-3$
$\Rightarrow \sum n(A\cap B)=12$

$\therefore n(A\cup B\cup C)=52-12=40$
$\therefore n\left(E_0\right)=50-40=10$