[f. as prec. + -(I)AL.] A. adj.
1. Math. (See quots. and B a.)
1817. Penny Cycl., IX. 155. The term factorial expression has been in some instances applied to an expression of which the factors are in arithmetical progression.
1860. Boole, Calc. Fin. Diff., 6. The term in which they [‘factorials’ in Boole’s sense] are involved is called a factorial term.
1867. Galbraith, Algebra (ed. 3), 396. Factorial functions … If the function consist of equi-different factors.
2. Of or pertaining to a factor (sense 4).
1881. Blackie, Lay Serm. v., 177. Large farms and factorial management have formed together an unholy alliance.
B. sb. Math. [In F. factorielle.]
a. gen. The product of a series of factors in arithmetical progression. In later usage sometimes with wider sense: The product of a series of factors which are similar functions of a variable that changes by a constant difference in passing from any factor to the next. Cf. FACULTY 2 c.
(Boole, Calc. Fin. Diff., 6, defines the word as meaning each of the factors composing such a product, and uses factorial term for what other writers call a factorial.)
1816. Herschel, in Lacroix’s Diff. & Int. Calc., App. § 370. Any quantity of the form a xn + b xn–1 + &c. may be resolved into factorials by the method of indeterminate coefficients.
1845. Penny Cycl., Suppl. I., 559. Arbogast … proposed to call the different cases of xna by the name of factorials.
1867. Galbraith, Algebra (ed. 3), 397. To find the difference of a factorial.
b. spec. The product of an integer multiplied into all the lower integers.
For example, the factorial of 6 (written symbolically |6 or 6!) is 6×5×4×3×2 = 720. Factorials are of frequent use in the investigation of Permutations and Probabilities.
1869. J. J. Raven, Ch. Bells Camb. (1881), 77. The number of changes that can he rung on a peal of bells is the factorial of the number of the bells.
