“The king of the south.” This king of the south is Ptolemy IV Philopator (reigned ca. 221–204 BC).
“the king of the north.” This is Antiochus III the Great (reigned ca. 223–187 BC). Both Ptolemy and Antiochus amassed huge armies. “According to Polybius, Ptolemy’s forces consisted of 70,000 infantry, 5,000 cavalry, and 73 elephants; whereas Antiochus’s army had 62,000 infantry, 6,000 cavalry, and 102 elephants” (Stephen Miller, The New American Commentary: Daniel). Despite the fact that the armies were closely matched, Ptolemy IV Philopator won a decisive victory and basically wiped out the northern army, with Antiochus barely escaping into the desert.
“who will set forth a great multitude.” The text is “and he will,” and it is referring to the king of the north. The king of the north will set forth a multitude, that is, Antiochus III will be the agressor; however, that multitude will be given into the hand of the king of the south, Ptolemy IV Philopator, who will defeat Antiochus.