How many ways can you
arrange a pack of cards?

  • Let's start with one card, we'll call it A. We can only arrange it one way.
  • Now we have two cards A and B, and we can arrnage them in two ways: AB or BA.
  • With three cards A, B and C there are six different ways: ABC, BAC, ACB, CAB, BCA, CBA
Here's how to work out the numbers involved. To start with, for one card it's 1. For two cards we work out 2x1=2 and for three cards it's 3x2x1=6. In maths we can write 3x2x1 as 3! The ! is called the factorial sign. Many calculators have a dedicated ! button.

Now we'll bring in a few more cards.

  • Four cards can be arranged in 4! = 24 ways.
  • Five cards can be arranged in 5! = 120 ways.
  • Six cards can be arranged in 6! = 720 ways.
Hold on tight! Here come the BIG numbers...
You can arrange 13 cards
in over 6 billion ways!
  • Ten cards can be arranged in 10! = 3,628,800 ways
  • Thirteen cards can be arranged in 13! = 6,227,020,800 ways
  • Twenty cards can be arranged in 20! = 2,432,902,008,176,640,000. (That's nearly two and a half billion billion ways.)

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So how many different ways could we arrange all the 52 cards in a pack?

Simple! We just need to work out 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 about the same as an 8 with 67 zeros after it.

Exactly how big is this number? Let's have a think about it.
The dinosaurs died out 65,000,000 years ago, and the earth itself is 4,500,000,000 years old. Four and a half billion years seems like a long time, yes?
Now let's suppose that all the people on Earth started to arrange packs of cards at the rate of one pack, per person, 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 possible arrangements!
So you can be pretty sure that every time you shuffle a pack of cards, you are creating an arrangement that has never existed before. You're making history!


There are more stats and facts about games here:

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Chances of Card Hands

The Betting Guide - and how to work out the Bookie's profit!

The Roulette Wheel ... and how to worry the casino!

The One-Armed Bandit Slot Machine

How many ways can you put a pack of cards in order?

The National Lottery Odds

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