By 332 bce which regions had fallen to alexanders army

The correct answer is Damascus, Tyre, Gaza, Egypt.

In 322 BCE, Alexander the Great had conquered Damascus, Tyre, Gaza, and Egypt.

Alexander The Great or Alexander III of Macedon (356 BC 323 BC) was the King of Macedon in Greece. In 322 BCE, his powerful Army had conquered Damascus, Tyre, Gaza, and Egypt.  

In Syria, Alexander commanded his Army through Cappadocia until Mount Taurus. He defeated the Persians in the Battle of Granicus and in the Battle of Issus. Then, Alexander conquered the most important city of Phoenecia, Tyrus on the Mediterranean coast. The conquest went on and he invaded Gaza when he entered the well defended fortified city on the hill. Alexander kept on his march until he reached Egypt where the Egyptians offered the control of Egypt and made him Pharaoh and considered him the incarnation of the God RA.