How is 7 factorial 5040

7 factorial (7!) equals 5040 by multiplying all positive integers from 1 through 7 sequentially: 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040. You can verify this step-by-step: 1 × 2 = 2, then 2 × 3 = 6, then 6 × 4 = 24, then 24 × 5 = 120, then 120 × 6 = 720, and finally 720 × 7 = 5040.

This particular factorial value appears frequently in mathematics and has interesting properties: 5040 is highly composite (divisible by every integer from 1 through 10 except 9), making it historically significant—Plato even suggested in "Laws" that an ideal city should have 5040 citizens because of the number's exceptional divisibility. In practical terms, 7! represents the number of ways to arrange 7 distinct items in a sequence, which equals 5040 different permutations. This calculation appears in probability problems (like determining possible arrangements of 7 people in a line), in puzzle-solving (calculating possible configurations), and in algorithm analysis where factorial complexity describes computational growth rates. The number's many factors make it useful in scheduling problems and modular arithmetic applications.