8. (Average/Difficult) Suppose each of five sticks is broken into a long part and a short part. The ten parts are arranged into five pairs and glued back together so that again there are five sticks. What is the probability that each long part will be paired with a short part?
Answer: $\frac{8}{63}$
Solution: The 5 short pieces and 5 long pieces can be lined up in a row in $\frac{10!}{5!5!}$ ways. Consider each of the 5 pairs of consecutive pieces as defining the reconstructed sticks. Each of those pairs could combine a short piece $(S)$ and a long piece $(L)$ in two ways: $S L$ or $L S$. Therefore, the number of permutations that would produce 5 sticks, each having a short and a long component is $2^{5}$, and so the desired probability is $2^{5} /\binom{10}{5}=\frac{8}{63}$.