[Combinatorics]
Problem: A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$.

Solution: Let $x$ be the number of orange marbles. Then the probability of drawing an orange marble is $\frac{x}{x+20+12}=\frac{x}{x+32}$. If this probability equals $\frac{1}{y}$, then $y=\frac{x+32}{x}=1+\frac{32}{x}$. This expression represents an integer only when $x$ is a factor of 32 , thus $x \in\{1,2,4,8,16,32\}$. The corresponding $y$-values are $33,17,9,5,3$, and 2 , and their sum is $\mathbf{6 9}$.

Final answer: 69