Theorem 1.2

For integers $k$ and $n$ such that $0\le k\le n$, the total number of $k$-combinations of a set of $n$ objects is $C(n,k)=\frac{n!}{k!(n-k)!}$

Proof

Since a set of $k$ elements ($k$-permutations) can be arranged into exactly $k!$ sequences,
$P(n,k)=C(n,k)k!$
i.e.
In the $k$-combinations of $n$, order doesn’t matter
$\therefore$ To ‘make order matter’ you can use the Multiplication Principle to order the set of $k$-combinations into the set of $k$-permutations