Definition of "factorial"

The expression n C r {\displaystyle {}_{n}C_{r}} means the number of combinations of n {\displaystyle n} things taken r {\displaystyle r} at a time. It is also written ( n r ) {\displaystyle {\binom {n}{r}}} , and is equal to n ( n − 1 ) ( n − 2 ) ( n − 3 ) ⋯ ⋯ ( n − r + 1 ) r ! {\displaystyle {\frac {n(n-1)(n-2)(n-3)\cdots \cdots (n-r+1)}{r!}}} or n ! / r ! ( n − r ) ! {\displaystyle n!/r!(n-r)!} , in which r ! {\displaystyle r!} is read "factorial r {\displaystyle r} " and denotes the product of all the integral numbers from 1 {\displaystyle 1} to r {\displaystyle r} inclusive.

1916 July, M. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, page 428

This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly as ∏ k = 1 n k = 1.2.3 ⋯ n . {\displaystyle \prod _{k=1}^{n}k=1.2.3\cdots n.} But what does this equal?Well, of course we can say that it is equal to n {\displaystyle n} factorial and write ∏ k = 1 n k = n ! {\displaystyle \prod _{k=1}^{n}k=n!} and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!

2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School‎, volume 47, number 4, page 40

I shall denote, as usual, the factorial product of the numbers up to n {\displaystyle n} by n ! {\displaystyle n!} , but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently define n ! ! = n . ( n − 2 ) ! ! . {\displaystyle n!!=n.(n-2)!!.} Starting with 1 ! ! = 1 , 2 ! ! = 2 , {\displaystyle 1!!=1,\quad 2!!=2,} it follows that, for positive values of n {\displaystyle n} , n ! ! = n ⋅ n − 2 ⋅ n − 4 ⋯ , {\displaystyle n!!=n\cdot n-2\cdot n-4\cdots ,} where the last factor is either 1 {\displaystyle 1} or 2 {\displaystyle 2} , according as n {\displaystyle n} is odd or even.

1903, Arthur Schuster, “On some Definite Integrals, and a New Method of Reducing a Function of Spherical Co-ordintes to a Series of Spherical Harmonics”, in Philosophical Transactions of the Royal Society A, volume 200, page 182

The definition of addition is an example of definition by induction, sometimes called recursive definition. As another example of this, consider the following inductive definition of the factorial function.We define n ! {\displaystyle n!} for n ∈ N {\displaystyle n\in \mathbb {N} } by the following two properties,(i) 1 ! {\displaystyle 1!} is defined to be 1 {\displaystyle 1} ,(ii) for all k ∈ N {\displaystyle k\in \mathbb {N} } , we define ( k + 1 ) ! {\displaystyle (k+1)!} to be ( k + 1 ) × k ! {\displaystyle (k+1)\times k!} .

1995, K. E. Hirst, Numbers, Sequences and Series, Butterworth-Heinemann, page 21

The latter sold the goods to a customer who was cashier to certain creditors of the agents without disclosing the factorial capacity in which they acted.

2004, The Digest: Annotated British, Commonwealth, and European Cases