﻿ Proof of number of way to arrange a pack of cards
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# Proof that there are more ways of arranging a pack of cards than there have been seconds since the universe began

There are 52 cards in a standard pack of playing cards.
The total number of ways of arranging such a pack of cards=52!

52! ~ 8.06582 x 1067

The Univesre is thought to be between 10 and 16 billion (16,000,000,000) years old. That is 16 billion years since The Big Bang.
Let us assume the highest value of 16 x 109 years.

There are 60 seconds in a minute.
There are 60 minutes in an hour.
There are 24 hours in a day.
There are 365 days in a year (or 366 in a leap year).

Therefore, the largest reasonable estimate for the number of seconds since the Universe began can be calculated as: 60 x 60 x 24 x 366 x 16,000,000,000 = 505958400000000000 seconds ~ 5.1 x 1017

8.06582 x 1067 > 5.1 x 1017 (by a factor of 1.59 x 1050)

Therefore it can clearly be seen that there are many more ways of arranging a standard set of playing cards than there have been seconds since the Universe began. Moreover, this will continue to be the case long after our own Sun and every other star in the Universe has run out of hydrogen and faded away approximately one hundred trillion years (1014) after the Big Bang, when the Universe will still be less than 3.2 x 1021seconds old.

In fact, remarkably, there are even more ways of arranging half a pack of playing cards than there have been seconds since the Universe began, as 26! ~ 4.03 x 1026 > 5.1 x 1017