The maximum number of nodes at level \( k \) in a binary tree is \( 2^k \). This means each level can hold at most double the nodes of the preceding level, beginning with 1 node at level 0 (the root). For instance, level 1 has 2 nodes, level 2 has 4 nodes, and so forth.
- \( k^2 \) (A) is incorrect; the number of nodes in a binary tree does not grow quadratically with the level number.
- \( 2k \) (C) is also incorrect; the node count increases exponentially with \( k \), not linearly.
- \( k! \) (D) is incorrect as the factorial function does not represent the number of nodes per level in a binary tree.
Therefore, the correct answer is (B), \( 2^k \).