Cardinality of Set of Subsets

Theorem
Let $$S$$ be a set such that $$\left|{S}\right| = n$$.

Then if $$m \le n$$, the number of subsets $$T$$ of $$S$$ such that $$\left|{T}\right| = m$$ is $$\frac {n!} {m! \left({n - m}\right)!}$$

Proof
For each $$X \subseteq \N_n$$ and $$Y \subseteq S$$, let $$B \left({X, Y}\right)$$ be the set of all bijections from $$X$$ onto $$Y$$.

Let $$\mathbb{S}$$ be the set of all subsets of $$S$$ with $$m$$ elements.

By Cardinality of Power Set and Proper Subset has Fewer Elements, $$\mathbb{S}$$ is finite, so let $$s = \left|{\mathbb{S}}\right|$$.

Let $$\beta: B \left({\N_n, S}\right) \to \mathbb{S}$$ be the mapping defined as $$\forall f \in B \left({\N_n, S}\right): \beta \left({f}\right) = f \left({\N_m}\right)$$.

For each $$Y \in \mathbb{S}$$, the mapping:
 * $$\Phi_Y: \beta^{-1} \left({Y}\right) \to B \left({\N_m, Y}\right) \times B \left({\N_n - \N_m, S - Y}\right)$$

defined as:
 * $$\Phi_Y \left({f}\right) = \left({f_{\N_m}, f_{\N_n - \N_m}}\right)$$

is also (clearly) a bijection.

By Cardinality of Set of Bijections, $$\left|{B \left({\N_m, Y}\right)}\right| = m!$$ and $$\left|{B \left({\N_n - \N_m, S - Y}\right)}\right| = \left({n - m}\right)!$$.

So by Cardinality of Cartesian Product, $$\left|{\beta^{-1} \left({Y}\right)}\right| = m! \left({n - m}\right)!$$.

It is clear that $$\left\{{\beta^{-1} \left({Y}\right): Y \in \mathbb{S}}\right\}$$ is a partition of $$B \left({\N_n, S}\right)$$.

Therefore by Number of Elements in Partition, $$\left|{B \left({\N_n, S}\right)}\right| = m! \left({n - m}\right)! s$$.

Consequently, as $$\left|{B \left({\N_n, S}\right)}\right| = n!$$ by Cardinality of Set of Bijections, it follows that $$m! \left({n - m}\right)! s = n!$$ and the result follows.

Comment
This expression crops so often in mathematics it has been given a special notation.

For all $$m, n \in \N$$, the number of subsets of $$m$$ elements of a set having $$n$$ elements is defined as:



\forall m, n \in \N: \binom n m = \begin{cases} \frac {n!} {m! \left({n - m}\right)!} & : m \le n \\ 0 & : m > n \end{cases} $$

The number $$\binom n m$$ is known as a binomial coefficient.

The reason for this is apparent from the Binomial Theorem.