Question 36
Let $n$ be a positive integer. A sequence $(a_{0}, \ldots, a_{n})$ of integers is acceptable if it satisfies the following conditions:
a. $0 = |a_{0}| \lt |a_{1}| \lt \cdots \lt |a_{n-1}| \lt |a_{n}|$.
b. The sets $\{ |a_{1} - a_{0}|, |a_{2} - a_{1}|, \ldots, |a_{n} - a_{n-1}| \}$ and $\{ 1, 3, 9, \ldots, 3^{n-1} \}$ are equal.
Prove that the number of acceptable sequences of integers is $(n+1)!$.
a. $0 = |a_{0}| \lt |a_{1}| \lt \cdots \lt |a_{n-1}| \lt |a_{n}|$.
b. The sets $\{ |a_{1} - a_{0}|, |a_{2} - a_{1}|, \ldots, |a_{n} - a_{n-1}| \}$ and $\{ 1, 3, 9, \ldots, 3^{n-1} \}$ are equal.
Prove that the number of acceptable sequences of integers is $(n+1)!$.