The number n! is the product 1 × 2 × 3 × 4 ×... × n, that is, the product of all the natural numbers from 1 up to n, including n itself where 1 is a natural number. It is called either "n factorial" or "factorial n." Thus 5! is the number 1 × 2 × 3 × 4 × 5, or 120.
Older books sometimes used the symbol In for n factorial, but the numeral followed by an exclamation point is currently the standard symbol.
Factorials show up in many formulas of statistics, probability, combinatorics, calculus, algebra, and elsewhere. For example, the formula for the number of permutations of n things, taken n at a time, is simply n!. If a singer chooses eight songs for his or her concert, these songs can be presented in 8!, or 40,320 different orders. Similarly the number of combinations of n things r at a time is n! divided by the product r!(n - r)!. Thus the number of different bridge hands that can be dealt is 52! divided by 13!39!. This happens to be a very large number.
When used in conjunction with other operations, as in the formula for combinations, the factorial function takes precedence over addition, subtraction, negation, multiplication, and division unless parentheses are used to indicate otherwise. Thus in the expression r!(n - r)!, the subtraction is done first because of the parentheses; then r! and (r - n)! are computed; then the results are multiplied.
As n! has been defined, 0! makes no sense. However, in many formulas, such as the one above, 0! can occur. If one uses this formula to compute the number of combinations of 6 things 6 at a time, the formula gives 6! divided by 6!0!. To make formulas like this work, mathematicians have decided to give 0! the value 1. When this is done, one gets 6!/6!, or 1, which is, of course, exactly the number of ways in which one can choose all six things.
As one substitutes increasingly large values for n, the value of n! increases very fast. Ten factorial is more than three million, and 70! is beyond the capacity of even those calculators which can represent numbers in scientific notation.
This is not necessarily a disadvantage. In the series representation of sine x, which is x/1! - x3/3! + x5/5! -..., the denominators get large so fast that very few terms of the series are needed to compute a good decimal approximation for a particular value of sine x.