# Factorial sequences

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

Zurtex
Homework Helper
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

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? 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?

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Zurtex
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misskitty said:
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? 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:

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

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?

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.

I think that makes sense.

Sidenote: why does 0! equal zero?

~Kitty

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.

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HallsofIvy
Homework Helper
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
$$\frac{7!}{4!3!}\frac{1}{2^7}$$.

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

dextercioby
Homework Helper
misskitty said:
I think that makes sense.

Sidenote: why does 0! equal zero?

~Kitty

By definition

$$0!=:1$$

Daniel.

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

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dextercioby said:
By definition

$$0!=:1$$

Daniel.

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

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

~Kitty

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

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

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

Hmm, I'm getting a little lost here. ~Kitty

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.

misskitty said:
Sidenote: why does 0! equal zero?
Another typical answer is that factorials are a specific case of the gamma function, and gamma(0) is 1

Zurtex
Homework Helper
JonF said:
Another typical answer is that factorials are a specific case of the gamma function, and gamma(0) is 1
Gamma(0) is not defined. Gamma is not defined on all non-positive integers.

However Gamma(1) = 0!

matt grime
Homework Helper
You ask why 0! should be one, but surely it is just as valid a question to ask why 1! is 1 or n! is defined to be n*(n-1)*(n-1)...*1.

Lots of mathematical objects are simply defined since they have been proven to be useful, ie we find ourselves using something a lot and realize it is worthwhile fixing the notation. The good ones become universal.

Factorials (possibly) came about from studying combinations of choosing and arranging objects. In this sense n! has a nice meaning for n>0, and it is convenient to have 0!=1 for many reasons as have been mentioned. It is not directly deducible from the definition of n! as n(n-1)..1 but there is nothing to stop us extending the definition and the choice of 0!=1 means that we can simply declare that 0!=1 and n!=n*(n-1)!, and to many people *this* is the definition of n! for n=>0 and it so happens that when n>0 this is the number of ways of arranging n objects and is equal to n(n-1)...1. I forgot who it was who posted it, but (perhaps unintuitively) we say there is one way to arrange 0 objects too.

This is very common: we start of with a simple definition of something, perhaps motivated by real life and then we mathematically refine it 'til we have the "best" definition. Best means here "is very general, ie applies to a lot of situations" but "not too general as to be practically useless".

I suppose one might say that given some definition there are two things we may do: deduce things from it or extend it. Here we can think that we have extended. The important thing is that the extension is consistent with the original and it is.

Here, for instance is an attempt to extend the factorial to (positive) rational numbers that doesn't work:

let p/q be a fraction with both p and q positive. Define (p/q)! to be p!/q!. This isn't consitent with the notion of factorial. Why? well, we would require that (1/2)! and (2/4)! were the same since 1/2 and 2/4 are the same rational number. but

(1/2)!=1/2 and (2/4)!=2/24=1/12.

hello there

some of the applications would be mainly statistics- like probability distributions- hyper-geometric distributions, binomial distributions, poisson distribution, gamma and beta distributions etc all these are a part of a theory that models reality - statistical analysis, which had originated from combinatorics and permutations

steven