Probability of being dealt four-of-a-kind in a set of $5$ cards? 
You are dealt a hand of five cards from a standard deck of playing cards. Find the probability of being dealt a hand consisting of four-of-a-kind. 

If possible, please provide a hint first before the answer.
One of the first things that came to me was $\dfrac{52}{52} \cdot \dfrac{51}{51} \cdot \dfrac{3}{50} \cdot \dfrac{2}{49} \cdot \dfrac{1}{48}$ but this was of course wrong.
Then, I realized that the third card could be the same as the first OR second card, so I tried $\dfrac{52}{52} \cdot \dfrac{51}{51} \cdot \dfrac{7}{50} \cdot \dfrac{2}{49} \cdot \dfrac{1}{48}$, which is also wrong. 
Then I realized that we probably need to add up the probabilities of situations where the first card is the one that doesn't match, or the second is the one that doesn't match, etc. I think this method leads to the solution, but I don't think we're intended to solve it this way. I think the solution looks something like $\dfrac{x}{ {{52}\choose{5}}}$, but I'm not sure what should be in the numerator.

The simplest explanation might be the following:
there are ${52}\choose{4}$ possible combinations of 4 cards in a deck of 52. Then, with 5 cards, you can have 13 * 5 possible four of a kind. Divide the latter by the former.  
