# The factorial

The factorial
10.3
As a further application of the multiplication principle, let us count the number of permutations of a sequence containing distinct terms, like 1, 2, 3, n. Well, let us form a permutation a1, an of the sequence 1n. Well, we can do it in n steps. In the first step, which is a1, we have n choices. Now, for each shoe choice of the first element, we have n minus 1 choices for the second and so on, up to the step n, where we have just one choice. Can you apply the multiplication principle 1 from the final sequence a1, an. We know that a1 was chosen a step 1, a2 was choosing a step 2, an was choosing step n.
63.7
Whenever we have a sequence as a final result, we can apply the multiplication principle if the elements of the sequence correspond to the outcomes of the various steps. So in particular, the application of the multiplication principle gives us n times n minus 1 times times 2 times 1 elements. Possible permutations. This number is called the factorial of n.
96.4
Now, the factorial of n is defined even when any is equal to 0. For convenience, the factorial of n is the product of the terms from n to 1 when n is greater or equal than 1, and it is equal to 1 when n is equal to 0. The factorial often can be approximated in term of exponentials and powers. More precisely, we have the Stirling formula, which is the factorial of n is asymptotic to square root of 2pi n times n to the n times e 2 minus n. e is the [INAUDIBLE] number. This means that the quotient of the two sequences turns to 1 as n turns to plus infinity.
148.4
So as an application, we can approximate number of digits of the factorial of 100. Well, you know the number of digits of a number equals the floor of the logarithm in the 10 basis of the number plus 1. So let us approximate the logarithm of the factorial of 100. Well, if we are probably the rulers of the fact of the logarithm, we get something like 157.0 something, so the floor of the logarithm of 100 is 157. When we add 1, we obtain 158 digits, and actually turns out that the factorial of 100 has got 158 digits. The factorial grows very quickly.

The factorial of a natural (n) is the natural number [n!=ntimes (n-1)cdotstimes 2times 1] The factorial appears in several combinatorial formulas: why?