[Math] the practical application of factorials

factorial

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

Best Answer

$$\sin x=\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\cdots$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\cdots$$ $$e^x=1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots$$ 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.

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