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.

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answer is B.

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Detailed Solution

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 $M_{r}^{'}$ and the set of days when there were clear afternoon is $A_{r}^{'} .$
By condition (b), we get $M_{r} \cap A_{r} = ϕ,$
By (d), we get $M_{r}^{'} = 6,$
By (c), we get $A_{r}^{'} = 5,$
and by (a), we get $M_{r} \cup A_{r} = 7.$
$M_{r}$ and $A_{r}$ are disjoint sets and $n (M_{r}) = d - 6, n (A_{r}) = d - 5.$
$∴$ Applying the principle of inclusion and exclusion, we get
$n (M_{r} \cup A_{r}) = n (M_{r}) + n (A_{r}) - n (M_{r} \cap A_{r})$