TITLE: Understanding when to use product rule and when to use sum rule?
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: As Arturo showed, you can use both rules depending:

choose the woman ($n$ ways), then choose the man ($n-1$ ways) this is a sequence so you multiply to get $n(n-1)$. They are independent because you removed the dependency by subtracting 1.
choose all possible pairs $n^2$ (using the product rule), then exclude the pairs not wanted $n$, or $n^2 - n$. The latter is just an inverse of the rule of sum, where the total possible pairs is the sum of the given couples $n$ plus all the other ways of making the couples (exercise for the reader...or rather the next problem in the sequence).