# Factorial sequences

1. Jul 17, 2005

### misskitty

Hey there everyone,

We were discussing factorial sequences in my last pre-calculus class. Factorials are pretty cool. I asked if they had any rel world applications or examples I could put into my notes. She then told us if we could find an example that we'd get extra credit on our quiz, I'm interested in sticking it in my notes since I missed the chance on that quiz anyway (unless she changed it at the last minute).

Here's the problem. I've been googling to find some examples and all I'm getting are explanations of what they are. I don't need to know what they are, I need to find an example. Google isn't helping. Does anyone know of a real life example of factorials? I really need help.

~Kitty

2. Jul 17, 2005

### Zurtex

Well the factorial function is very useful for algebra because it saves on having to write more than mathematicians need to (mathematicians in general are lazy when it comes to writing). Hmm, real world applications though, I imagine a very similar function like the gamma function probability has a lot, but factorial is a bit difficult as it only works on non-negative integers.

Here is a good page to look at I suppose: http://en.wikipedia.org/wiki/Factorial

3. Jul 17, 2005

### misskitty

I checked that site actually. It was pretty good. I'll agree with you there. I was just curious if there we any tangible examples. For example; we use bacteria as a real life example of exponantial growth right? Is there anything like that for factorials?

4. Jul 17, 2005

### robert Ihnot

Well, would you consider the trigomentric functions,

$$sinX=\frac{X^0}{0!}-\frac{X^3}{3!} +\frac{X^5}{5!}+..(-1)^n\frac{X^n}{n!}...$$

Or the log, or the value of e^x, all of which can be computed by factorial expressions?

Last edited: Jul 17, 2005
5. Jul 17, 2005

### Zurtex

The factorial function eventually grows faster than any exponentials. Generally speaking we use factorials as they help us in maths more than anything.

A number called e, very useful in exponentials, is often defined by:

$$e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!} + \ldots$$

Nicely it works out when you have e to the power of x:

$$e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \ldots + \frac{x^n}{n!} + \ldots$$

6. Jul 17, 2005

### misskitty

I remember e^x from algebra. I didn't know we could use them in trig functions though. So there is no tangible example for factorials other than how we use them in higher math. What about pascal's triangle?

7. Jul 17, 2005

### robert Ihnot

A good approximation of e is easy to get since we just sum up: 1+1+1/2+1/6+1/24+1/120......in each case dividing the previous denominator by n for the nth term (starting with the 0th term, and yes! 0! is defined as 1). Since 10! =3,628,800, and 1/10! =2.76x10^(-7), we don't have to go very far to get a useful approximation.

8. Jul 17, 2005

### misskitty

I think that makes sense.

Sidenote: why does 0! equal zero?

~Kitty

9. Jul 17, 2005

### robert Ihnot

Another way factorials are used is in probability problems. The number of ways that K things can be taken from N things is the binominal coefficient:

$$\frac{N!}{K!(N-K)!}$$

An example of this is given four things: a,b,c,d, how many ways can we take them 1 at a time? WEll, that's just 4: a, b,c,d. And is represented by 4!/(3!1!) = 24/6 = 4.

A more difficult question is, how many ways can we take two things at a time from four? That is 4!/(2!2!) = 24/4=6. Which can be demonistrated by chosing them: ab,ac,ad,bc,bd,cd.

If we wanted to take 50 things from 100, well, we'd surely want to use the formula rather than do the counting:

$$\frac{100!}{50!50!}$$, which is approximately 1.01x10^29.

Last edited: Jul 17, 2005
10. Jul 17, 2005

### HallsofIvy

Staff Emeritus
The number of ways in which you can put 6 books on a shelf is 6!

The probability of flipping 4 heads and three tails in 7 flips with a fair coin is
$$\frac{7!}{4!3!}\frac{1}{2^7}$$.

11. Jul 17, 2005

### misskitty

That is a good one Halls. I didn't think about that one. Thanks.

So factorials are used in combinetrics, but when I read up one it there wasn't a good explanation of what they are used for.

~Kitty

12. Jul 17, 2005

### dextercioby

By definition

$$0!=:1$$

Daniel.

13. Jul 17, 2005

### robert Ihnot

misskitty: Sidenote: why does 0! equal zero?

That is a difficult question, in a sense, but generally the answer is that it is a definition which is convienent and useful. See http://www.themathpage.com/aPreCalc/factorial.htm

Here is what mathworld says, The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).
http://mathworld.wolfram.com/Factorial.html

Last edited: Jul 17, 2005
14. Jul 17, 2005

### misskitty

I know 0! equals 1 by definition, I was asking why. Good to hear from you Daniel.

~Kitty

15. Jul 17, 2005

### misskitty

The page is good Robert, but it doesn't explain why.

~Kitty

16. Jul 17, 2005

### robert Ihnot

I'm kind of slow at this and added more material on 0!: Here is what mathworld says, The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).
http://mathworld.wolfram.com/Factorial.html

--------------------------------------------------------------------------------

17. Jul 17, 2005

### robert Ihnot

We might also deduce it from taking N things N at a time:

$$\frac{N!}{N!0!}=1.$$

18. Jul 17, 2005

### misskitty

Hmm, I'm getting a little lost here.

~Kitty

19. Jul 17, 2005

### LittleWolf

Hi, maybe this will help. Using n!=n*(n-1)! consider this sequence

3!=3*2!
2!=2*1!
1!=1*0!

Using the last sentence, the left hand side has 1! which equals 1 and the right hand side has the product of 1 * something we call 0!. So 0!=1.

20. Jul 18, 2005

### JonF

Another typical answer is that factorials are a specific case of the gamma function, and gamma(0) is 1