Textbook
For an integer $n\ge1$, the factorial is $n!=1\cdot2\cdots n$; a separate definition sets $0!:=1$. For $n\ge1$, it satisfies $n!=n(n-1)!$. The exclamation mark denotes the factorial operation.
There are $n!$ ways to arrange $n$ distinct objects. The first position has $n$ candidates; after choosing one, the next has $n-1$, then $n-2$, down to one. Multiplying the choices gives the factorial. If objects are indistinguishable, this unadjusted calculation counts identical arrangements repeatedly.
The recurrence matches the counting interpretation: choosing the first object leaves $(n-1)!$ orders for the rest, and there are $n$ possible first choices. At $n=1$, it reads $1!=1\cdot0!$, so $0!=1$ preserves one consistent rule. In this definition the argument is a nonnegative integer; inserting a negative number into the finite chain does not automatically define a factorial.
Definitions and further reading: Mathematics for Computer Science.
In plain language
A factorial counts orders of distinct objects: four tasks offer four first choices, then three, two, and one. Each schedule still contains four tasks; twenty-four counts the ways to arrange them. A fifth task multiplies the number of orders by five through the extra branching choices. An empty list has one order, so $0!=1$: even no tasks receive a valid plan, without stretching the working day.
Example
The example arranges four illustrative cards A, B, C, and D. The scene fixes one possible choice at a time, while the formula counts every branch, not just the displayed arrangement.
Factorials: Counting Arrangements
First position
Any of the four cards may come first.
Step 1 of 3: First position
Read all steps
First position
Any of the four cards may come first.
- State
- AcurrentBcurrentCcurrentDcurrent
Continuations
After A, there are three choices and then two.
- State
- A | BcurrentA | CcurrentA | Dcurrent
Complete arrangement
The last card is forced; there are 24 arrangements.
- State
- A | B | C | Dcurrent24current
The scene follows one branch, such as A first, but that branch still permits six tails made from B, C, and D. Four such groups of six correspond to the four first choices. This gives $4\cdot6=24$, matching the full formula. Adding $4+3+2+1$ would count options at individual decisions without combining them, so it would not count completed arrangements.
Related concepts: Choosing without order.