$p^m (1-p)^{n-m}$ is the probability of winning $m$ out of $n$ games in a specific order, ie. e.g. win, then lose, then lose, then win, then win... To get the probability for any order you multiply by ${n \choose m}$ which is exactly what you have done.
You are correct!
$p^m (1-p)^{n-m}$ is the probability of winning $m$ out of $n$ games in a specific order, ie. e.g. win, then lose, then lose, then win, then win... To get the probability for any order you multiply by ${n \choose m}$ which is exactly what you have done.