A puzzle book I saw posed a question similar to the following.
Suppose an urn has a total of $N$ balls. $B$ are blue and the rest are red. The probably of choosing $5$ blue balls in a row (without replacement) is exactly $\frac{1}{2}$. What are the possible values of $N$ and $B$?
In other words, what are the positive integer solutions to:
$$ \prod_{i=0}^{4}\frac{B-i}{N-i}=\frac{1}{2} $$
One solution is $N=10,B=9$. My question is, are there other positive integer solutions? Is this related to a well-known family of equations?
Here is what I have tried.
For any $x$ we can solve for $N$ from the equation: $$\prod_{i=0}^{4}\frac{\left(N-x\right)-i}{N-i}=\frac{1}{2}$$
Call this solution $N_{x}$. If $N_{x}$ is an integer, then $N_{x},\left(N_{x}-x\right)$ is a solution. Since $N$ and $B$ are integers, $B=N-x$ for some integer $x$. Thus, this process will find all integer solutions.
I tried this in Mathematica for x from 1 to 10000 and $x=1$ (the $N=10,B=9$ solution above) was the only $x$ with an integer $N_{x}$.This also tells us there is no solution with $N\leq 77252$ since $N_{10000} \approx 77252.2$.
I used this:
Table[Solve[Product[(n - x - i)/(n - i), {i, 0, 4}] == 1/2, Integers], {x, 1, 10000}]
