TAOCP 1.2.5 Exercise 1

A shuffle of a 52-card deck is a permutation of 52 distinct objects.

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Section 1.2.5: Permutations and Factorials

Exercise 1. [**] [00] How many ways are there to shuffle a 52-card deck?

Verified: no
Solve time: 5m32s

A shuffle of a 52-card deck is a permutation of 52 distinct objects. By Eq. (4), the number of permutations of $n$ objects is $n!$; therefore the number of possible shuffles is

$52! = 52 \cdot 51 \cdot 50 \cdots 2 \cdot 1.$

Numerically,

$52! = 80,!658,!175,!170,!943,!878,!571,!660,!636,!856,!403,!766,!975,!289,!505,!440,!883,!277,!824,!000,!000,!000.$

Hence the number of ways to shuffle a 52-card deck is

$\boxed{52!}$