40320

40320 is the natural number that comes after 40319 and before 40321. In mathematics, it is written as 8! and equals the product 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8. The factorial notation indicates the product of all positive integers up to a given number, and 8! is a classic example used to illustrate permutation counts.

Prime factorization: 40320 = 2^7 × 3^2 × 5 × 7.

Combinatorial significance: It is the number of permutations of eight distinct objects, i.e., the size of the symmetric group S8. It also represents the number of ways to arrange eight labeled items in order.

Time and units: 40320 seconds equals 11 hours 12 minutes; 40320 minutes equals 28 days.

Historical note: The factorial notation '!' was introduced by French mathematician Christian Kramp in 1808. Since then factorials have become a fundamental tool in combinatorics, probability, and mathematical analysis.

In broader use, 40320 serves as a standard example of factorial growth, illustrating how rapidly the number increases with the addition of each factor in the product. It also appears in computations involving permutations, arrangements, and certain counting problems across mathematics.