Final answer:
The Americans won the Battle of New Orleans on January 8, 1815, after the Treaty of Ghent was signed on December 24, 1814, but before it was ratified by the American Government in February 1815. The correct option is 2.
Step-by-step explanation:
The Battle of New Orleans was won by the Americans under the command of Andrew Jackson on January 8, 1815. This battle was the last major conflict of the War of 1812, occurring after the Treaty of Ghent was signed.
The Treaty of Ghent, which officially ended the war, was signed on December 24, 1814.
However, due to slow communication at the time, the treaty was not ratified by the American Government until February 1815, which is why the Battle of New Orleans took place after the treaty had been signed.