> 
> Given a traditional deck of $52$ cards, find the probability that a
>  $5$-card hand contains four cards of the same rank.
> 
> 
> 

So far, we only dealt with the notion of a finite sample space and that each elementary event is equally probable. So I defined the sample space 

$$ \Omega := \{\omega : \omega = (a\_1,a\_2,a\_3,a\_4,a\_5), a\_j \neq a\_k, j\neq k, a\_i \in \{1,\dots,52\}\}$$ since the sampling is without replacement. Now I am trying to figure out $$|\{\text{exactly four cards of the same rank}\}|$$ since then we have $$P(\{\text{exactly four cards of the same rank}\}) = \frac{|\{\text{exactly four cards of the same rank}\}|}{|\Omega|}$$ Does anyone has a nice way of calculating the cardinality of the above set? I tried to split it up into several disjoint subsets, but I do think this gets quite complicated.