What Is N Factorial Used For?
Date: 08/29/2001 at 18:02:22 From: Stuart Reed Subject: What is n factorial used for? Can you please tell me what n! or n factorial is used for. I know what it is (5! is 120, because 1*2*3*4*5 = 120), but not what it is used for. I have heard that it is used in probability, but what for, and how? How does it work, and what are its uses?
Date: 08/30/2001 at 10:49:46 From: Doctor Ian Subject: Re: What is n factorial used for? Hi Stuart, The factorial function is useful in computing the number of combinations or permutations that can be constructed from a set of objects. You can read about these uses in our FAQ: Permutations and Combinations http://mathforum.org/dr.math/faq/faq.comb.perm.html Very briefly, here is the kind of problem that would give rise to the factorial function. Suppose you have three people (Alan, Bob, Cindy), and you want to know how many ways they can stand in line. For a small number like 3, the answer is easy to construct by trial and error: Alan, Bob, Cindy Alan, Cindy, Bob Bob, Alan, Cindy Bob, Cindy, Alan Cindy, Alan, Bob Cindy, Bob, Alan But for larger numbers of people, it becomes very difficult to avoid duplicating some sequences, or skipping some by accident. With a little thought, we can see that if we have N people, then there are N choices for who might go into the first position: 1. Alan 2. Bob 3. Cindy 4. Diane Having made that first choice, there are N-1 choices for who might go in the second position, for _each_ of the original choices: 1. Alan 1. Bob 2. Cindy 3. Diane 2. Bob 1. Alan 2. Cindy 3. Diane 3. Cindy 1. Alan 2. Bob 3. Diane 4. Diane 1. Alan 2. Bob 3. Cindy And there are N-2 choices for who might go into the third position, again for _each_ of the previous sets of choices: 1. Alan 1. Bob 1. Cindy 2. Diane 2. Cindy 1. Bob 2. Diane 3. Diane 1. Bob 2. Cindy 2. Bob 1. Alan etc. 2. Cindy 3. Diane 3. Cindy 1. Alan 2. Bob 3. Diane 4. Diane 1. Alan 2. Bob 3. Cindy I'll leave the rest of the chart for you to fill in, if you wish. Note that the final number of possible orderings of N objects will be ways to make the next choice | v N * (N-1) * (N-2) * ... * 1 ^ ^ | | ways to ways to make the make the first choice final choice which is exactly the factorial function. When dealing with combinations and permutations (and, by extension, when dealing with probabilities), this kind of reasoning comes up over and over again, which is why the factorial function is so important. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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