Ordered pairs in a shuffle

What this ordered-pairs probability question tests

This is a medium-difficulty probability question that asks you to reason about constraints on a random permutation. Specifically, it requires you to find the probability that a shuffled sequence of 100 numbers satisfies a particular ordering property across all 50 consecutive pairs.

To solve problems like this, candidates must recognize the underlying structure—in this case, how many permutations satisfy the given condition versus the total number of permutations. The key insight is often that you do not need to enumerate outcomes explicitly; instead, you identify a symmetry or combinatorial principle that pins down the probability directly. Firms use questions in this family to test whether you can move fluidly between counting arguments, conditional reasoning, and probability fundamentals.

• Symmetry and equiprobability in permutations
• Counting valid configurations
• Independence of pairwise constraints
• Powers of 2 in probability (a common signal)