Editing Falling and rising factorials (section)

==Examples and combinatorial interpretation==
The first few falling factorials are as follows:

<math display="block">
\begin{alignat}{2}
(x)_0 & &&= 1 \\
(x)_1 & &&= x \\
(x)_2 &= x(x-1) &&= x^2-x \\
(x)_3 &= x(x-1)(x-2) &&= x^3-3x^2+2x \\
(x)_4 &= x(x-1)(x-2)(x-3) &&= x^4-6x^3+11x^2-6x
\end{alignat}</math>

The first few rising factorials are as follows:

<math display="block">
\begin{alignat}{2}
x^{(0)} & &&= 1 \\
x^{(1)} & &&= x \\
x^{(2)} &= x(x+1) &&=x^2+x \\
x^{(3)} &= x(x+1)(x+2) &&=x^3+3x^2+2x \\
x^{(4)} &= x(x+1)(x+2)(x+3) &&=x^4+6x^3+11x^2+6x
\end{alignat}</math>

The coefficients that appear in the expansions are [[Stirling numbers of the first kind]] (see below).

When the variable <math>x</math> is a positive integer, the number <math>(x)_n</math> is equal to the number of [[k-permutation|{{mvar|n}}-permutations from a set of {{mvar|x}} items]], that is, the number of ways of choosing an ordered list of length {{mvar|n}} consisting of distinct elements drawn from a collection of size <math>x</math>. For example, <math>(8)_3 = 8\times7\times6 = 336</math> is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. On the other hand, <math>x^{(n)}</math> is "the number of ways to arrange <math>n</math> flags on <math>x</math> flagpoles",<ref name=Feller>
{{cite book
 |first=William |last=Feller
 |title=An Introduction to Probability Theory and Its Applications
 |volume=1 |at=Ch. 2
}}
</ref>
where all flags must be used and each flagpole can have any number of flags.  Equivalently, this is the number of ways to partition a set of size <math>n</math> (the flags) into <math>x</math> distinguishable parts (the poles), with a linear order on the elements assigned to each part (the order of the flags on a given pole).