The factorial can be neatly expressed in Eindhoven notation (a lot more ASCII-friendly than the pi notation): n! = (* : (i ∈ Z) ∧ (0 < i ≤ n) : i).
Among other things, the factorial of n is the number of different ways n items can be ordered. For example, take 3 items, call them a, b and c. These can be arranged in 6 different ways, abc, acb, bac, bca, cab, or cba; 3! = 6. 20 items can be ordered in 20! different ways, etc..
Logically, this makes perfect sense. When choosing the first item, there are n to choose from. When choosing the second, a totally independent choice, there are (n - 1) items to choose from. Just which (n - 1) items in particular depend on the first choice, but the number of items left for the second choice is always the same. Because the two choices are independent, the numbers are multiplied. For the third choice, there are (n - 2) items to choose from, etc., and for the last choice there is only 1 item to choose from. So the total number of ways to order them are (n * (n - 1) * (n - 2) … * 1) = n!.