Question

A factory has $80$ workers and $3$ machines. Each worker knows to operate at least two machines. If there are $65$ persons who know to operate machine $\text{I},$ $60$ for machine $\text{II}$ and $55$ for machine $\text{III},$ then the minimum number of persons who know to operate all the three machines is

Answer 1

Let $x$ workers know to operate all the machines.


Also, $\alpha +\beta +x=65$
$\alpha +\gamma +x=60$
$\beta +\gamma +x=55$
$\Rightarrow 2(\alpha +\beta +\gamma )+3x=180\cdots \left(1\right)$
And $\alpha +\beta +\gamma +x=80\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$x=20$