I'm a bit confused as to when you should use the Incl/Excl principle. Take for example the following problem:
How many 13-card hands can be selected from the standard 52-card deck with exactly four spades or exactly four diamonds?
My university tutor solved this problem by using the principle. My theory would be that you need to use it in this case because a 13-card hand with exactly four spades can also contain four diamonds, and a hand with exactly four diamond can also contain four spades - so you end up counting too many times.
Is there a trick or a general rule for when you should use it?
Sub-question: My tutor wrote this for choosing exactly four spades, and I believe there might be a mistake:
$c(13,4) * c(9,39)$
$c(13,4)$ is the number of ways to choose four positions for the spades
$c(9,39)$ is the number of ways to choose the remaining 9 cards that are not spades
But isn't it missing another $c(13,4)$ to choose the four spades out of 13 spades?