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The term 'n factorial,' denoted as n!, represents the product of every positive integer from 1 up to n. This mathematical operation is fundamental in combinatorics and mathematics in general, particularly for calculating permutations and combinations.

For example, if n is 5, then 5! (5 factorial) would be calculated as 5 × 4 × 3 × 2 × 1, which equals 120. This product continuously increases as n increases, demonstrating its role in counting the total arrangements of n distinct objects.

In contrast, the other choices provide incorrect interpretations. The sum of all integers from 1 to n describes a different mathematical concept, which would rather be represented by the formula n(n + 1)/2. The ratio of n to the next integer does not have any relevance to the concept of factorials, and the square of n (n^2) is simply n multiplied by itself, which is also unrelated. Therefore, 'n factorial' definitively is the product of every number between 1 and n.