A student on vacation for d days observed that (a) it rained seven times in the morning or afternoon; (b) when it rained in the afternoon, it was clear in the morning; (c) there were five clear afternoons and (d) there were six clear mornings. Find d.

Let the set of days it rained in the morning be $M_r$ and the set of days it rained in the afternoon be $A_r.$ Then, clearly the set of days when there were clear morning is $M_r^{\text{'}}$ and the set of days when there were clear afternoon is $A_r^{\text{'}}.$  
By condition (b), we get $M_r\cap A_r=\varphi ,$  
By (d), we get $M_r^{\text{'}}=6,$ 
By (c), we get $A_r^{\text{'}}=5,$  
and by (a), we get $M_r\cup A_r=7.$  
$M_r$ and $A_r$ are disjoint sets and $n\left(M_r\right)=d-6,n\left(A_r\right)=d-5.$  
$\therefore$ Applying the principle of inclusion and exclusion, we get
$n(M_r\cup A_r)=n\left(M_r\right)+n\left(A_r\right)-n(M_r\cap A_r)$

$\Rightarrow 7=(d-6)+(d-5)-0\Rightarrow 2d=18\Rightarrow d=9.$