 There are 52 cards in a standard deck, as you know, and you've probably played many card games. You've shuffled decks to mix them, and you've tried to predict what cards have been distributed to other players. Hidden in that deck of cards are some amazingly huge numbers. Let's ask the following question:  This innocent-seeming question has an astounding answer. Let's work it out.  Imagine selecting cards from the deck and laying them out one by one.How many choices do you have for the first card you pick? 52, of course. When you choose the second card, you now have only 51 possibilities, since you've already chosen one and laid it out. Similarly your choice for the third card will be made from the 50 cards that are left in the deck. The number of possibilities for each card continues to decrease by one each time, until you get to the last card, for which there is only one possible way to lay a card. The total number of ways you can order the 52 cards is the product of all the ways you can lay each card. This is of course the Fundamental Counting Principle you learn about in Math 30. The total number of ways of laying out the deck is then: This is '52 factorial', or 52! (the exclamation point being the symbol for factorial) The value of 52! is incredibly large. It's a 68-digit number: 
You may be wondering how this answer can be worked out, since a calculator shows only 10 digits. Calculating 52! on a TI83, either by using the factorial function (in the Math menu) or by laboriously multiplying 52 x 51 x 50 x ... x 3 x 2 x 1, gives an answer in scientific notation, and you won't get more than the first 10 digits. In fact, 13! is the biggest number a calculator will handle before running out of display space and reverting to scientific notation.In order to get the actual 68 digits, you need to use a computer. The Windows calculator will show only the first 32 digits, and isn't powerful enough ... a special program is needed to hold the result. We cheated and looked it up! In scientific notation, and rounded, the number of different ways to shuffle a deck is: 8.1 x 1067 Just how large is this number anyway? What can we compare it to? Let's look at money first. There are about 7 billion people on Earth. Imagine that every single person was given one million dollars in pennies. A million dollars in pennies works out to 100,000,000 cents. The total number of pennies distributed to all the people in the world would then be 100,000,000 x 7,000,000,000 or 700,000,000,000,000,000 pennies. In scientific notation this is 7 x 1017 cents. A lot of pennies! But this number is still approximately 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times smaller than the number of ways to deal a deck of cards! How about this. The size of the universe is estimated to be something like 15 billion light years in radius. A light year is the distance light travels in a year, or about 140,000,000,000,000,000,000,000 kilometres. The volume of the universe is then (assuming it's spherical) about 1.1 x 1070 cubic kilometres. This number is only about 100 times the number that represents the number of ways you can lay out a deck of cards. Saying this a different way, it would take a whole 1% of the entire universe to lay out all the different arrangements of the deck, if you put each one in its own cubic kilometre.  Let's look finally at a comparison with cards themselves. Imagine you could shuffle the deck 1000 times per second. Everyone on Earth (all 7 billion people) has their own deck of cards, and they're all shuffling them too, 1000 times per second. Imagine everyone continues to do this for the next 10 billion years. In all those shuffles you won't have rearranged the cards even a small fraction of the total number of possible ways!  What this means, in practical terms, is that every time you shuffle a deck of 52 cards, you get a new arrangement of the cards that has never been seen by anyone before, and will never occur again! That's a lot of power in one little deck of cards! |