Lower Bound for Comparison Based Sorting

Let there be \(n\) elements. There are \(n!\) ways they can be arranged. Consider a decision tree where on every node we compare two elements. Every time we make a comparison we have two results, so this is a binary tree. The leaves must be \(n!\) since every possible arrangement of the \(n\) elements must be reachable. So the longest path is \(\log_{2}(n!) = \Omega(n\log_{2}n)\)