What is the practical application of factorials

I'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer.

I understand that a factorial of n items gives you the number of ways you can arrange the given items.
For example: If there are two coins - you can arrange them in two different ways - like wise if you have 3 coins - there are 6 ways you can arrange them.

I've the following questions:

1. What are the other applications than arranging number of items.
2. I'm reading about the gamma function to be used for finding factorials of non-integers (decimals). What is the use of this - as we would never tend to arrange like 2.5 (two-and-a-half) items.
3. I plotted the following range of numbers with its factorial using gnuplot (software). I want to know how this software interpolates the value of the factorial between two integers. Because I have not told gnuplot to use the gamma function

number  factorial
0           1
1           1
2           2
3           6
4           24
5           120

plot "factorial" using 1:2 smooth bezier

I've very limited knowledge in mathematics - say school level - I'd prefer a simple answer which can be related to easily

$$\sin x=\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\cdots$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\cdots$$ $$e^x=1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots$$ The sine and cosine functions are important in trigonometry, which has practical applications to surveying and astronomy. The exponential function is used for the calculation of compound interest.

1. During a mathematical education program you will usually encounter it in calculus, for example Taylor's theorem $$ f(x) = \sum_{k=0}^\infty \frac{f^{(n)}(x_0)}{k!}(x-x_0)^k. $$ and the binomial theorem $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}, \quad \binom{n}{k} = \frac{n!}{k! (n-k)!} $$ or combinatorics (art of counting). Permutations show up in algebra. On this site my last use of factorials and gamma function was this (at first look rather frightning) equation: \begin{align} \frac{(-n)^{n-1} \Gamma(n+1)}{(1-n)_{n-1}} &=\frac{(-n)^{n-1} n!} {(1-n)(1-n+1)(1-n+2)\cdots -2 \cdot -1} &=\prod_{k=1}^{n-1} \frac{(k+1) n^2}{n^2-kn} \\ &=\frac{2 n^2}{n^2- n}\cdot\frac{3 n^2}{n^2-2 n}\cdot\frac{4 n^2}{n^2-3 n} \cdots \frac{n^3-3n^2}{4n} \cdot \frac{n^3- 2n^2}{3 n}\cdot\frac{n^3- n^2}{2 n}\cdot n^2 \\ &= n^n \end{align} Historically gambling problems were a major reason for the development of combinatorics and probability theory.
2. It is a valid question to extend the factorial, a function with natural numbers as argument, to larger domains, like real or complex numbers. The gamma function also showed up several times as certain integrals, so mathematicians gave it a name and of course noted the relationship to factorials. See the graph at the end of this posting. My favourite application of the gamma function is the volume and surface of a ball in $n$ dimensions: $$ V_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}r^n \quad\quad S_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} r^{n-1} $$
3. You ordered that interpolation via "smooth bezier". A Bézier curve is an interpolation function. Drop that part or try different plotting options, see "help plot" within gnuplot. For example:

   plot "factorial" using 1:2 with linespoints

Here is a plot together with the gamma function, or to be more precise, $\Gamma(x+1)$:

Well, although the question title is asking for practical applications the OP is really asking for "real world" applications (perhaps "practical" should be replaced by "pragmatic" here). If so, one area springs to mind: Gambling.

In any card game, if you want to calculate (or even estimate) the probability of favorable outcomes, you have to have a working knowledge of factorials.

Think of any video game, or a track meet relay, where you pick players to go first, second, third or fourth in a race. Or anytime you have the capability to do a number of acrivities in any order, like a chore list.

I am using Mario and Sonic at the Olympic Games as an example. You can put Sonic first, or second or third or fourth. You can arrange Sonic and any other person on your team to come up with the best team to win the race.

The number of possibilities depends on what total number of teammates you have. In this case you start with 4 slots to fill on your team. You multiply each remaining number underneath 4 until you get to 1 then stop. By multiplying you find the total number of combinations you could dwell on before you begin a game. So in this example $4!$ is $4\times3\times2\times1=24$ possibilties.

If making a chore list of 12 items you would find, that $12!$ is $12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1=479,001,600$ possibilties. This is not practically useful but shows the power of possibilities.

$4!= 24$. So 24 hours in a day. The order you spend your time is a factorial you use everday without thinking.

$3!=6$. Anything that has 6 unique flavors, you can combine in any order, or a game with 6 possible moves you could do in any order is another factorial.

Hope this example helps. I am a 4th/5th grade math teacher and use this example to challenge my students to learn more math and apply it!