I would like to prove that we need at least five comparisons to find median amoung five numbers.
And my proposition:
Let's consider tree of comparisons. There are at least $5 \cdot {4\choose 2} = 30$ possiblities, so in our tree we have at least $30$ leaves. So what is minimal height of this tree when we have at least $30$ leaves. (height is number of comparisons).
So $$height \ge \log_2(30) > 4$$
Hence, $height \ge 5$.
Is it correct proof ?