Answer by Sarthak Chatterjee:

*Every time you shuffle a deck of cards, it is likely that you've come up with an arrangement that has never been seen before in human history.*

Consider a standard deck of 52 playing cards. Choose one card and keep it aside. How many choices do you have? Well, there are 52 cards you can select 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 says that you can do both of these in [math] 52 \times 51 [/math] ways. If you continue this procedure, you have [math] 52 \times 51 \times 50 \times \cdots \times 2 \times 1 [/math] 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.

[math] 52!=80658175170943878571660636856403766975289505440883277824000000000000 \approx 8.0658 \times 10^{67} [/math]

Now how many shuffles have been made in human history? This is well nigh 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.)

[math] 7 \: \mathrm{billion} \times 700 \times 365 \times 24 \times 60 \times 60 \approx 1.546 \times 10^{20} [/math]

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

[math] \displaystyle \frac{1.546 \times 10^{20}}{8.0658 \times 10^{67}} \approx 1.9 \times 10^{-48} [/math]

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.

Source: MW – Shuffling Cards

Image Courtesy: DeviantArt

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