Cardinality of Set of Injections/Informal Proof

Theorem
Let $S$ and $T$ be sets.

The number of injections from $S$ to $T$, where $\left|{S}\right| = m, \left|{T}\right| = n$ is often denoted $P_{nm}$, and is:


 * $ {}^m P_n = \begin{cases}

\dfrac {n!} {\left({n - m}\right)!} & : m \le n \\ 0 & : m > n \end{cases}$

Informal Proof
This is the same question as determining how many permutations there are of $m$ objects out of $n$.

Informally, the thinking goes like this.

We pick the elements of $S$ in any arbitrary order, and assign them in turn to an element of $T$.

The first element of $S$ can be mapped to any element of $T$, so there are $n$ options for the first element.

The second element of $S$, once we've mapped the first, can be mapped to any of the remaining $n-1$ elements of $T$, so there are $n-1$ options for that one.

And so on, to the $m$th element of $S$, which has $n - \left({m-1}\right)$ possible elements in $T$ that it can be mapped to.

Each mapping is independent of the choices made for all the other mappings, so the total number of mappings from $S$ to $T$ is:


 * $\displaystyle n \left({n-1}\right) \left({n-2}\right) \ldots \left({n-m+1}\right) = \frac {n!} {\left({n - m}\right)!}$