Question

If $n(A\cap B)=10,n(B\cap C)=20$ and $n(A\cap C)=30$, then the greatest possible value of $n(A\cap B\cap C)$ is___

Answer 1

Given, $n(A\cap B)=10,n(B\cap C)=20$ and
$n(A\cap C)=30$.

$A\cap B\subset (B\cap C)$
And

$A\cap B\subset (A\cap C)$
Therefore,

Max$\left\{n\right(A\cap B\cap C\left)\right\}=min\left\{n\right(A\cap B),n(B\cap C),n(A\cap C\left)\right\}$

= 10
Hence,
The greatest possible value of $n(A\cap B\cap C)$ is 10.