My friend was asked the following problem in an interview a while back, and it has a nice answer, leading me to believe that there is an equally nice solution.
Suppose that there are 42 bags, labeled $0$ though $41$. Bag $i$ contains $i$ red balls and $42-(i+1)$ blue balls. Suppose that you pick a bag, then pull out three balls without replacement. What is the probability that all 3 balls are the same color?
The problem can be solved easily by using some basic identities with binomial coefficients, and the answer is $1/2$. Moreover, if $42$ is replaced by $n$, the answer does not change, assuming $n>3$. However, this computational approach obscures any hidden structure there might be. Ideally, I would like a simple and direct proof that the probability of getting RRR is the same as the probability of getting RBB.
So, is there a nice solution to the problem, one that could be explained fully to someone without the use of paper? Or is there no good way to explain this beyond computational coincidence?