Confusion about the proof of If $n = 2^k - 1$ for $k \in \mathbb{N}$, then every entry in row $n$ of pascal's triangle is odd.

Question

I saw there were existing posts for this problem (If $n = 2^k - 1$ for $k \in \mathbb{N}$, then every entry in row $n$ of pascal's triangle is odd.), but I'm still confused. This problem is in the Book of Proofs, under the section of contrapositive proofs. What is a contrapositive or direct proof for this problem?