Carlota likes playing games at the local arcade. She has 15 credits to spend, and wants to spend it all playing Accuracy Ball (which costs 1 credit per round) or Dance Fever (which costs 2 credits per round). In how many ways can Carlota spend all 15 credits at the arcade, if she wants to play at least three rounds of each game? (The order in which Carlota plays games matters: Playing Accuracy Ball, then Dance Fever is different from playing Dance Fever, then Accuracy Ball.)

So i thought 15-9 credits to get 6 credits and then arrange them too but also order does matter so im confused.

Guest Aug 29, 2022

#1**0 **

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...

BuilderBoi Aug 29, 2022