How Many 5-Card Hands of Poker Are There?

There are 52 cards in a regular poker deck. If 5 cards are dealt at random, how many possibilities are there?

Questions like this require the formula for combinations. In a combination, we don't care about the order of the elements. In poker, if you are dealt the ace of spades, the king of hearts, the 10 of diamonds, the 2 of clubs, and 3 of clubs, that's the same as being dealt the 3 of clubs, the 2 of clubs, the 10 of diamonds, the king of hearts and the ace of spades. 

When the order is important, we must use the formula for permutations. 

Both formulas have something called factorials. 3! (read as 3 factorial) means 3*2*1, so 3! = 6. 

If we want to know the number of ways of picking r elements from a set of n elements, we use:

n!/(r!(n-r)!)

For the case of drawing 5 cards from a deck of 52:

52!/(5!(52-5)!) = 52!/(5!(47!)) = 2,598,960

This means that 2,598,960 people could play poker at the same time and all have a different hand. The catch is that it would require 49,980 decks of cards.