The following diagram shows a piece of steel placed inside a solenoid. If the switch, S, is closed and opened several times, the end A of the steel bar becomes ...

When the switch S is closed, current flows through the solenoid. Using the Right-Hand Grip Rule (or Right-Hand Solenoid Rule), if you curl the fingers of your right hand in the direction of the current flow (upwards on the front of the coils as shown by the arrows), your thumb will point to the right. This indicates that the right end of the solenoid becomes the magnetic north pole.

Since end A of the steel bar is at the right end of the solenoid, it will be induced with a north pole.

The core material is a "piece of steel". Steel is a hard magnetic material, meaning it is difficult to magnetize but retains its magnetism once magnetized. The process of closing and opening the switch several times repeatedly exposes the steel to the magnetic field, effectively magnetizing it.

Because steel is a hard magnetic material and it has been repeatedly magnetized, it will become a permanent magnet. Therefore, end A will become a permanent north pole.

The correct option is B.
$\boxed{\text{B}}$