Final answer:
The Republicans won the presidency as a third party in 1860 with Abraham Lincoln, making them the only third party to ever win the presidency.
Step-by-step explanation:
The answer to the question of which third-party candidate has ever won the presidency is c) the Republicans in 1860. Prior to the election of 1860, the Democratic Party splintered into multiple factions, leaving the relatively new Republican Party an opening to win the presidency. The Republican Party, formed by anti-slavery activists and individuals who defected from other parties like the Whigs and Democrats, won its first presidential election with Abraham Lincoln at the helm. Thus, the Republicans were the only third party to successfully break the two-party dominance and win the presidency up until that point in history.