Given a list of distinct numbers (we can assume this because this is a worst-case analysis), there are n {\displaystyle n} factorial permutations exactly one of which is the list in sorted order. The sort algorithm must gain enough information from the comparisons to identify the correct permutation. If the algorithm always completes after at most f ( n ) {\displaystyle f(n)} steps, it cannot distinguish more than 2 f ( n ) {\displaystyle 2^{f(n)}} cases because the keys are distinct and each comparison has only two possible outcomes.