The boisterous crowd ____ with its merry-making even amidst pouring rain.

The phrasal verb 'carried on' means to continue an activity despite obstacles or interruptions. The crowd continued its merry-making even though it was raining, so 'carried on' fits perfectly. 'Carried out' means to perform or complete a task, 'played on' usually refers to continuing to play (more apt for games/music), and 'flowed on' is unidiomatic here. Therefore option (c) is correct.