Order does not matter, repetition not allowed

How many ways can we select $k$ persons from a group of $n$ people if the order of selection does not matter?

• As seen before, if the order of selection matters, there are $\frac{n!}{(n-k)!}$ ways.
• But then for each set $A$ of $k$ people, we counted the the people in $A$ the number of ways $k$ persons can be ordered. ($k!$)
• So, if the selection order does not matter, then the number of ways is: $$\begin{array}{rl}\frac{\frac{n!}{(n-k)!}}{k!}& =\frac{n!}{(n-k)!\cdot k!}\\ & =\left(\begin{array}{c}n\\ k\end{array}\right)\text{ or }{}_n{C}_k\end{array}$$ This leaves us with the binomial coefficient equation.