How many different ways can you
put a pack of cards in order?
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You can arrange these 13 cards
in over 6 billion ways!
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This demonstration is taken from the book DO YOU FEEL LUCKY?
- Suppose you have 1 card: A. There's only one way you can put it in order.
- If you have two cards A and B, you can arrange them in two different ways. They can be AB or BA.
- With three cards A,B and C there are six different ways: ABC, ACB, BAC, BCA, CAB, CBA
The way the numbers arrive is that for one card it's 1. For two cards it's 2x1=2 and for three cards it's 3x2x1=6. The quick way to write 3x2x1 is 3! The ! sign is called a factorial. Some calculators (including the one found on computers) have a special ! button.
Let's move on a bit.
- If you've got four cards you can arrange them in 4! = 24 ways.
- If you've got five cards you can arrange them in 5! = 120 ways.
- If you've got six cards you can arrange them in 6! ways = 720 ways.
Get ready for the BIG numbers!
- The number of ways you can arrange 10 cards is 3,628,800
- The number of ways you can arrange 13 cards is 6,227,020,800
- The number of ways you can arrange 20 cards is 2,432,902,008,176,640,000. That's two and a half billion billion.
So how many ways can you order all the 52 cards in a pack?
The sum is 52x51x50x49x48....x4x3x2x1 and the answer is:
80,658,175,170,943,878,571,660,636, 856,403,766,975,289,505,440, 883,277,824,000,000,000,000
That's roughly the same as an 8 with 67 zeros after it.
To put this in perspective, the dinosaurs died out 65,000,000 years ago, and the age of the earth is just 4,500,000,000 years. Now suppose everybody in the world was to arrange packs of cards at the rate of one per second, it would take 600,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 years to get all the combinations! That's why you're VERY unlikely ever to shuffle a pack of cards the same way twice.
There are more card facts and big numbers at Chances of Card Hands.
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