Factorial is most simply defined as a function on positive integers. 5 factorial (written as ) means . In general then, for a positive integer , . For applications to combinatorics, it will also be useful to define .
is the number of possible orders for a set of objects. For example, if we arrange the letters , , and , here are all the options: You can see that there are possible orders for objects, and . Why does this work? We can prove this by induction. First, we'll see pretty easily that it works for object, and then we can show that if it works for objects, it will work for . Here's the case for object. Now we have the objects , and slots to put them in. If is in the first slot, now we're ordering remaining objects in remaining slots, and by our induction hypothesis, there are ways to do this. Now let's suppose is in the second slot. Any orderings that result from this will be completely unique from the orderings where was in the first slot. Again, there are remaining slots, and remaining objects to put in them, in an arbitrary order. There are another possible orderings. We can put in each slot, one by one, and generate another orderings, all of which are unique, and by the end, we will have every possible ordering. We know we haven't missed any because has to be somewhere. The total number of orderings we get is , which equals .
The factorial function can be defined in a different way so that it is defined for all real numbers (and in fact for complex numbers too).
We define as follows: where is the gamma function: Why does this correspond to the factorial function as defined previously? We can prove by induction that for all positive integers : First, we verify for the case where . Indeed: Now we suppose that the equality holds for a given : and try to prove that it holds for : We'll start with the induction hypothesis, and manipulate until we get the equality for . By the product rule of integration: This completes the proof by induction, and that's why we can define factorials in terms of the gamma function.