How many cards must be selected from a standard deck of $52$ cards to guarantee that at least three hearts are present among them?

A standard deck has

$13$ hearts

and

$52-13=39$ non-heart cards.

To guarantee at least $3$ hearts, consider the worst case.

First, we may select all $39$ non-heart cards.

Then, we may select only $2$ hearts.

So, cards selected without guaranteeing $3$ hearts:

$39+2=41$

Now, the next card must be a heart.

So, required number of cards is

$41+1=42$