What does the notation n! signify in mathematics?

The notation n! in mathematics represents the factorial of n, which is defined as the product of all positive integers from 1 up to n. For example, if n is 5, then 5! (5 factorial) is calculated as 5 × 4 × 3 × 2 × 1, resulting in 120. Factorials are commonly used in combinatorics, probability, and various areas of mathematics to calculate permutations and combinations.

Other interpretations of the options can cause confusion, as none accurately reflects the concept of factorial. The sum of numbers from 1 to n would correlate with a different mathematical series, while the highest exponent of n and the average of the first n integers are distinct concepts not related to factorials. Ultimately, understanding n! as the product of all positive integers up to n is essential, particularly for solving problems in areas involving counting and arrangements.