A deck of cards looks reassuringly small. There are only 52 cards to hold, sort and shuffle. Yet changing their order creates a mathematical problem so large that writing down every possible arrangement is hopeless.
The difference between an arrangement and a decision becomes easier to see in a game. Open a classic Klondike deal on PlaySolitaire, the free browser solitaire game, and look at the seven columns before moving anything. That layout gives us a practical example: the cards already have an order, but you cannot see all of it. Here are five facts that explain why that matters.
A deck has an astonishing number of possible orders
With the jokers removed, a standard deck contains 52 distinct cards. Any of them can come first. That leaves 51 possibilities for second place, then 50 for third, continuing until only one card remains.
Multiplying those choices gives 52 factorial, written 52!. The answer is roughly 8.07 multiplied by 10 to the power of 67, a number with 68 digits. The exclamation mark is mathematical notation, not a comment on how surprising the result is.
A smaller experiment makes the calculation tangible. Take an ace, a two and a three. There are three choices for the first card and two for the next, producing six orders altogether. Adding a fourth distinct card increases the number to 24. Each extra card does much more than add one new arrangement.
A tidy pattern is not automatically less likely
Imagine an ideal shuffle in which every complete order has the same chance. A deck arranged neatly by suit is then just as likely as one particular untidy sequence you write down beforehand.
The important words are “one particular”. There are many arrangements that look untidy, but relatively few that we would recognise as neatly organised. Comparing a single tidy arrangement with every untidy arrangement is a different question.
This also means appearance is a poor test of randomness. Two neighbouring kings do not, by themselves, prove that a shuffle was faulty. And a deck that looks thoroughly mixed does not establish that the shuffling method gave every possible order an equal chance.
Choosing two cards is different from ordering them
Suppose you pick two cards and place them side by side. There are 52 choices for the left position and 51 for the right, giving 2,652 ordered pairs.
Now suppose you care only about which two cards you picked. An ace followed by a queen represents the same selection as that queen followed by that ace. Every selection was counted twice, so dividing by two leaves 1,326 possible pairs.
This is the distinction between a permutation, where order matters, and a combination, where it does not. It explains why two questions about the same deck can produce different answers. Before calculating a probability, first decide exactly what you are counting.
Solitaire turns hidden order into a practical puzzle
Return to the Klondike layout. Its seven columns initially contain one, two, three, four, five, six and seven cards: 28 altogether. Only the top card of each column is exposed, leaving 21 hidden beneath them. The remaining 24 cards form the stock.
Those hidden cards are not waiting to acquire identities. They already belong to the deal; uncovering one changes what you know. That is why revealing a covered card can matter more than making a move that merely rearranges cards already visible.
The tableau also treats suits differently from the foundations. You build tableau sequences downwards in alternating colours, while each foundation grows upwards from ace to king in a single suit. A red six may fit on a black seven, but that alone says nothing about whether moving it helps uncover a useful card.
Replaying a deal is not the same as drawing a new one
A fresh deal changes the initial arrangement. Restarting the same deal lets you change your decisions while keeping that arrangement fixed. Remembering a previously hidden card gives you information you did not have on the first attempt.
That makes replaying useful for testing a specific idea. Did uncovering the longer column help? Did keeping a space for a king create another route? Compare the consequences rather than assuming that a quicker move was better.
None of this guarantees a win. It simply separates three things that are easy to confuse: the order of the cards, the information available to you, and the choices you make. An ordinary pack contains all three puzzles, even before you finish the first game.