I was doing some plotting on wolfram alpha and the following binomial
${n/2-n/4} \choose {n/4-5}$
was represented as:
$\frac{1}{480} ({\frac{n}{4}-4})({\frac{n}{4}-3})({\frac{n}{4}-2})({\frac{n}{4}-1})n$
I am curious if somebody can explain to me why this representation and what is the rationale behind it
We deal here with the notion of falling factorial.
Let us first rewrite your binomial coefficient in a simpler way, using classical formula: $\binom{N}{P}=\binom{N}{N-P}$:
$$\binom{n/4}{n/4-5}=\binom{n/4}{n/4-(n/4-5)}=\binom{n/4}{5}\tag{1}$$
A falling factorial is (indeed) defined as a polynomial with consecutive integer roots beginning at $0$ ; more precisely:
$$x_{(N)}:=x(x-1)(x-2) \cdots (x-N+1)\tag{2}$$
with the property:
$$\dfrac{x_{(N)}}{N!}=\binom{x}{N}\tag{3}$$
(this definition and this property can be found in the Wikipedia reference given above ; please note that this definition/property extends a property that we know as valid for integer values of $x$).
Setting $x=n/4, N=5$ (take care not to mix $n$ and $N$) and using (1),(2), and (3), it is easy to establish your identity.
Remark: you will see in the Wikipedia article the dual concept of rising factorial.
