Consider a standard deck of 52 playing cards. Choose one card from it and keep it aside.
How many choices you have? There are 52 cards and hence 52 choices.
Select another card. This time you have 51 choices, remember one card has been kept aside.
The fundamental principle of counting/probability says that you can do both of these in 52×51 ways. If you continue this way, you have 52×51×50×⋯×2×1 ways in which you can arrange your deck. The product of numbers from 52 to 1 is called the ‘factorial’ of 52, written 52! . This is an inconceivably large number.
Now let’s calculate how many shuffles have been made in human history? This is well impossible to calculate, but let’s take a shot.
Assume that 7 billion people (close to the current population of the world) have been shuffling cards once every second since 1300 AD (approximately 700 years), the time when the modern deck of cards appeared. The number of shuffles comes out be roughly (ignoring leap years etc.)
Of course, this is a gross overestimation, but it drives home the point. So, as a fraction of the total number of shuffles possible, the shuffles already done are a paltry
( 1.546×10^20 ) / ( 8.0658×10^67)≈ 1.9×10^-48
So next time you shuffle a deck of cards and lay them on the table, chances are you’re looking at some never-before-seen piece of history.