2. guy has a collection of 10 rulers, whose l...

# Explanation: ## Step1: Calculate total arrangements The total number of ways to arrange 10 rulers is \(n = 10!\). ## Step2: Calculate non - desired arrangements Use the principle of inclusion - exclusion. First, find the number of arrangements without consecutive blocks. Consider the "gaps" between non - consecutive rulers. But a more straightforward approach is to find the number of arrangements where the rulers in order sum to 1000. The lengths of the rulers are a geometric sequence \(a_n=2^{n - 1}\) (\(n = 1,2,\cdots,10\)) and \(\sum_{i = 0}^{9}2^{i}=\frac{1\times(1 - 2^{10})}{1-2}=1023\). Let's consider the "bad" arrangements (where the sum is not 1000). The number of arrangements where the sum is 1000 is non - trivial to count directly. However, we can use the fact that the total number of arrangements of 10 distinct objects is \(10! = 3628800\). The number of arrangements where the sum of the lengths of the rulers in order is not 1000 is very small compared to the total number of arrangements. In fact, there is exactly 1 way to arrange them to sum to 1000 (the correct ascending order \(1,2,4,\cdots,512\)). The probability \(P\) that the sum is 1000 (i.e., there is a consecutive block in the correct order) is \(\frac{1}{10!}\). # Answer: \(\frac{1}{3628800}\)