Q.In a survey of $100$ investors, $60$ hold shares and $45$ hold mutual funds; $25$ hold both. How many hold

Let $S$ = share-holders and $M$ = mutual-fund holders. "At least one" is $n(S \cup M)$, found by inclusion–exclusion; "neither" is the complement within the $100$ surveyed.

Let the universal set be the $100$ surveyed investors, $S$ the set holding shares and $M$ the set holding mutual funds. We are given $n(S) = 60$, $n(M) = 45$, $n(S \cap M) = 25$.

1. At least one. "Holds shares or mutual funds" is the union $S \cup M$. By the inclusion–exclusion formula,

   $n(S \cup M) = n(S) + n(M) - n(S \cap M) = 60 + 45 - 25 = 80.$

   So $80$ investors hold at least one of the two.
2. Neither. Those holding neither are outside $S \cup M$, i.e. the complement within the $100$: $n\big((S \cup M)'\big) = 100 - n(S \cup M) = 100 - 80 = 20.$ …

A Venn diagram with the overlap filled in first ($25$ in both), then the 'only' regions ($35$ and $20$), reaches the s …

The classic mistake is adding $60 + 45 = 105$ and forgetting to subtract the $25$ counted in both, which even exceeds the $100$ surveyed — a signal that the overlap …