There are 4 cases: He plays 3, 4, 5, or 6 rounds of Dance Fever.
For the first case, he plays 3 rounds of Dance Fever and 9 rounds of Accuracy Ball. There are \({12 \choose 3} = 220\) ways to place the Dance Fever, and the remaining spots are Accuracy Ball.
For the second case, he plays 4 rounds of Dance Fever and 7 rounds of Accuracy Ball. There are \({11 \choose 4} = 330\) ways to place the Dance Fever, and the remaining spots are Accuracy Ball.
For the third case, he plays 5 rounds of Dance Fever and 5 rounds of Accuracy Ball. There are \({10 \choose 5} = 252\) ways to place the Dance Fever, and the remaining spots are Accuracy Ball.
For the final case, he plays 6 rounds of Dance Fever and 3 rounds of Accuracy Ball. There are \({9 \choose 6} = 84\) ways to place the Dance Fever, and the remaining spots are Accuracy Ball.
So, there are \(220 + 330 + 252 + 84 = \color{brown}\boxed{886}\) ways.
I think...