Factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point. Thus, factorial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7.
The value of 0! is 1. The first reason why zero factorial is equal to one is that this is what the definition says it should be, which is a mathematically correct explanation (if a somewhat unsatisfying one). Still, one must remember that the definition of a factorial is the product of all integers equal to or less in value to the original number—in other words, a factorial is the number of combinations possible with numbers less than or equal to that number.
Because zero has no numbers less than it but is still in and of itself a number, there is but one possible combination of how that data set can be arranged: it cannot. This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! is equal to one because there is only a single possible arrangement of this data set.
There are n! different ways of arranging n distinct objects into a sequence, the permutations of those objects.
Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.
0 Comments