2024 Mock AMC 10 - Problem 21

Each time a Clash Royale chest is opened, a card is randomly selected (the chance of each distinct card being selected is equal) from set \mathcal{S}, which contains N \ge 2 distinct cards, for some positive integer N. Then, a copy of the selected card is given to the player (Set \mathcal{S} remains unchanged, and there is an ample supply of each card). Given that the cards “Hog Rider” and “Ram Rider” are in \mathcal{S}, and that the expected number of chests opened until a Hog Rider and a Ram Rider are drawn in that order is between 275 and 300, find the sum of the digits of N.

\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }14