Why Does a Shuffled Deck Almost Never Repeat an Order?

A standard deck contains 52 distinct cards. There are 52 choices for the first position, 51 for the second, and progressively fewer after that. The total number of possible orders is therefore 52 × 51 × … × 1, written as 52!. This is about 8.07 × 10^67. It does not prove that every shuffle creates something literally unprecedented in history; it shows that the space of possible arrangements is vastly larger than everyday intuition suggests.

Factorials grow quickly because each added object can be inserted into many positions within every earlier arrangement. Five cards have only 120 possible orders, while ten have 3,628,800. By the time we reach 52 cards, even checking a billion orders each second from the formation of the universe until now would cover only a tiny fraction.

However, “52! possible orders” does not mean a real shuffle produces each one with equal probability. Technique, repetition and human movement introduce biases; uniform randomness is a mathematical assumption. The restrained conclusion is that, when a shuffle is sufficiently close to random, exactly repeating one specified order is extraordinarily unlikely. The permutation count describes the size of the possibility space, not a guarantee that the physical process is perfectly random.

https://mathworld.wolfram.com/Factorial.html
https://www.ams.org/books/mbk/146/