factorielle

The mathematical concept of factorial — the product of all positive integers up to a given number — was used long before it had a name or a symbol. Hindu mathematicians explored permutations as early as the twelfth century. The Lilavati of Bhaskara II (1150 CE) included problems requiring factorial calculation. European mathematicians encountered the concept through combinatorics — counting the number of possible arrangements.

Christian Kramp coined the French term factorielle and introduced the exclamation mark notation (n!) in his 1808 work Elements d'Arithmetique Universelle. The choice of the exclamation mark was either whimsical or prescient — factorials produce numbers so large they seem to deserve exclamation. 10! = 3,628,800. 20! = 2,432,902,008,176,640,000. 100! has 158 digits. The growth is explosive.

Factorials are the mathematics of arrangement. How many ways can you shuffle a deck of 52 cards? 52!, which is approximately 8 × 10⁶⁷. That number is larger than the estimated number of atoms in the observable universe (~10⁸⁰, but still in the same general territory). Every time you shuffle a deck thoroughly, you almost certainly create an arrangement that has never existed before and will never exist again.

Stirling's approximation (1730) gave mathematicians a way to estimate large factorials without computing them directly: n! ≈ √(2πn)(n/e)ⁿ. The formula connects factorials to π and e — the two most important mathematical constants appearing in a formula about counting arrangements. Factorials link the discrete mathematics of combinatorics to the continuous mathematics of analysis. The exclamation mark earned its keep.