Fixed points of a permutation

Expected number of fixed points in a random permutation

This is an easy probability question that tests your ability to use linearity of expectation to solve a counting problem elegantly. Rather than enumerate permutations, you identify the right random variables to sum and compute their individual expectations.

The problem asks you to find the expected value of a quantity defined over a uniformly random permutation of a finite set. The key insight is to avoid brute-force enumeration and instead decompose the problem into simpler, independent pieces. This technique is foundational in probability and appears frequently in quant interviews because it separates clear thinking from computational struggle.

• Indicator random variables and their expectations
• Linearity of expectation (works even when variables are not independent)
• Symmetry arguments in combinatorics