When do we use factorials?

[Betrayer0fHope]

I won't be learning them for a while now, so I was wondering if anyone actually uses them for their work, or knows why someone would. Thanks.

[Roman]

Finding the number of combinations a system has and in probability stuff.

[camilus]

^Precisely.

But even more important is the theoretical uses of factorials, which helped us get some extremely important results in number theory. Such as:

e^x = \sum \limits_{n=0}^\infty {x^n \over n!}

replacing x with ix is where we are able to derive one of the most important results in all of pure mathematics, Euler's Formula: e^{ix} = \cos (x) +i\sin (x).

and that formula gives birth to the most beautiful identity in existence when letting x=\pi:

e^{i\pi} + 1 = 0

An equation connecting the fundamental numbers i, \pi, e, 1, 0, the operations of addition, multiplication, exponentiation, and the most important relation of all, =. Carl Gauss, one of the greatest mathematicians in history is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.

[Tom2]

Here's one from quantum mechanics. In the 1D harmonic oscillator problem we define the creation operator to be, in appropriately chosen units, a^{\dagger}=\frac{1}{\sqrt{2}}(x-ip). It's action on a harmonic oscillator eigenstate \psi_n is a^{\dagger}\psi_n=\sqrt{n+1}\psi_{n+1}, where n[/itex] is the energy level. That is, the operator "creates" an energy level.

Now observe the action of [tex]\left(a^{\dagger}\right)^k on the ground state \psi_0.

\left(a^{\dagger}\right)^k\psi_0
=\left(a^{\dagger}\right)^{k-1}\sqrt{1}\psi_1
=\left(a^{\dagger}\right)^{k-2}\sqrt{2\cdot 1}\psi_2
=\left(a^{\dagger}\right)^{k-3}\sqrt{3 \cdot 2\cdot 1}\psi_3
....etc

See the factors accumulating under the square root sign? That continues until we exhaust the k creation operators and end up with:

\left(a^{\dagger}\right)^k\psi_0=\sqrt{k!}\psi_k

[AlphaNumeric]

The Gamma fuction, which has the properties \Gamma(z+1) = z \Gamma (z) for complex z and for integer n, \Gamma(n) = (n-1)! comes up everywhere in quantum field theory, because you often have to do integrals involving spheres in n dimensions and there's an integral identity which currently escapes me (it's in Peskin and Schroder somewhere) which relates the volume of an n-sphere to n! and quantum field theory does the unpleasant thing of working in non-integer dimensions and so you get a generalised notion of factorial all over the place.

[iceaura]

I won't be learning them for a while now, At your level, everything in notation - factorials, square roots, exponents, logarithms, etc, - is basic and ordinary and will come up in almost any intellectual or professional field you may be drawn to later.

If you are posting on this forum, there is almost no chance that factorials will be useless to you in the future.

Plus, they shorthand and make easier some approaches to thinking and talking about things - for the pleasure of understanding.

The symbol is pronounced "shriek".

[funkstar]

The symbol is pronounced "shriek".
Huh. I've sometimes heard it pronounced "bang", but I think the most common is still to just say "factorial"...

[paulfr]

One major area is Combinations and Permutations.

The number of ways to select k items from n items is C[n,k] = n! / ( [n - k] ! k! )

The number of ways to arrange [or order] k items from n items [as in books on a shelf] is
P[n,k] = n! / ( [n - k] ! )