What are the chances of correctly guessing at least three of five cards? Picking five cards from a well-shuffled regular deck of cards (52 cards), the chances of correctly guessing all five card (in order) is one in $$\frac{52!}{(52-5)!} \quad \text{or} \quad \frac{52!}{47!}$$
But what are the chances of correctly guessing at least three cards correctly?
The guess is made before the cards are revealed.
 A: In this answer I am assuming that the player is obliged to make his guess of the 5 card sequence before any cards are revealed.  
Lets assume that the cards are  labeled 1,2,...,52 and that the player's ordered guess is 1,2,3,4,5 ( since this guess is equally likely to be correct as any other guess). 
What we want to count is the number of injective functions from {1,2,3,4,5} into {1,2,...,52} that have exactly k fixed points.  Here k will correspond to the exact number of cards guessed correctly by the player.  We must be familiar with inclusion/exclusion. 
Here is the Mathematica code that gives the probability that the player guesses exactly k cards correctly for k=0,1,2,3,4,5.

Table[Binomial[5, j] Sum[Binomial[5 - j, k] FactorialPower[52 - k - j, 5 - j - k] (-1)^k/FactorialPower[52, 5], {k, 0, 5 - j}], {j, 0, 5}]

The exact probabilities are:  $\frac{16649407}{18345600}, \frac{5541121}{62375040}, \frac{6511}{1834560}, \frac{2257}{31187520}, \frac{47}{62375040}, \frac{1}{311875200}$. 
