FTCE Professional Education Practice Exam

What does the notation '!' signify in factorials?

1. Sum of a sequence

2. Product of all positive integers up to a certain number

3. Random selection of integers

4. Average of a numerical set

The correct answer is: Product of all positive integers up to a certain number

The notation '!' in mathematics denotes the factorial of a non-negative integer. Specifically, the factorial of a number 'n', represented as 'n!', is defined as the product of all positive integers from 1 up to 'n'. This means that to calculate 'n!', you multiply together all whole numbers starting from 1 and going up to 'n'. For example, if n is 5, then 5! = 5 × 4 × 3 × 2 × 1 = 120. This concept is commonly used in areas such as combinatorics, probability, and algebra to calculate permutations and combinations, among other applications. The other options refer to different mathematical operations. The sum of a sequence involves adding numbers together, a random selection of integers could relate to probability and statistics but does not pertain to factorials, and the average of a numerical set involves calculating the mean, which also does not align with the definition of a factorial.