Suppose there are $n$ couples in a party. What is the way of choosing a man and a woman who are not a couple.
I can choose a woman in $n$ ways ($E_1$), and I am left with $n-1$ choices for a man not her husband ($E_2$). Now, I cannot decide if I should use the sum rule or the product rule, i.e are the total possibilities $2n-1$ or $n^2-n$. $E_1$ and $E_2$ do not seem to be independent, as the event E_1 automatically determines the set of $E_2$. The sum rule sounds possible, as $E_1$ and $E_2$ cannot occur simultaneously (because I will have to choose a woman first and then choose a man from the remaining set).
I am having trouble when to use sum and product rules in general.