From Matt Enlow comes an original puzzle of his that previously appeared in Math Horizons. (At first it may seem like there’s information missing, but I assure you that is not the case.)
A bag contains 100 marbles, and each marble is one of three different colors. If you were to draw three marbles at random, the probability that you would get one of each color is exactly 20 percent.
How many marbles of each color are in the bag?
Let $p$, $q$, and $r$ be the number of marbles of the three different colors.
We know that:
$$p + q + r = 100$$
and:
$$\frac{p}{100} \cdot \frac{q}{99} \cdot \frac{r}{98} \cdot 3! = \frac{1}{5}$$
$$=> pqr = \frac{100\cdot99\cdot98}{5\cdot6} = 32,340$$
From there, we can use prime factorization,
$$32,340 = 2 \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 7 \cdot 11$$
To achieve a sum of $100$,
$$21, 35, 44$$