There are $52$ people in a room. what is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 12$
Pretend you have $52$ people you want to place  in $12$ boxes, because there are $12$ months in a year. By the [Pigeonhole Principle](https://artofproblemsolving.com/wiki/index.php/Pigeonhole_Principle), one box must have at least $\left\lceil \frac{52}{12} \right\rceil$ people $\longrightarrow \boxed{\textbf{(D)} 5}$