What does factorial notation signify?

Factorial notation, denoted by an exclamation mark (n!), signifies the product of all positive integers from 1 to n. For example, if n is 5, then 5! equals 5 × 4 × 3 × 2 × 1, which equals 120. This notation is commonly used in permutations and combinations, where it helps calculate the number of ways to arrange or select objects.

The concept of factorial is foundational in combinatorics, particularly when determining the total number of arrangements of n distinct objects. Each integer contributes to the total product, leading to an exponential increase as n increases. This characteristic makes factorial notation very useful in probability and statistical calculations.

Other options do not accurately represent what factorial notation conveys. For instance, the sum of a sequence of numbers is not captured by factorials; instead, it denotes addition rather than multiplication. A special type of equation does not specifically pertain to factorials, and while counting the arrangements of objects relates to factorials, that is just one application of the broader definition that factorial notation provides. Thus, the product of all positive integers from 1 to n clearly encapsulates the essence of factorial notation.