Combinations: choosing objects

$n!$ represents the number of ways to rearrange $n$ distinct objects. That is, we will use all $n$ objects. There are situations where we don’t require all of them.

For example, consider three objects $A,B$ and $C.$ We may only require 2 of them. Listing the possibilities gives us $\{A,B\},\{A,C\}$ or $\{B,C\}$ so there are 3 ways to choose 2 objects from 3 if the order is not important (that is, $A,B$ and $B,A$ are considered the same choice in our example).

Listing is way too cumbersome for larger possibilities so we introduce a new symbol and formula to calculate such situations. For choosing 2 objects out of 3, we have ${\displaystyle \left(\genfrac{}{}{0px}{}{3}{2}\right)=3.}$

Technique: choosing $r$ objects out of $n$

There are $\left(\genfrac{}{}{0px}{}{n}{r}\right)$ ways to choose $r$ distinct objects out of $n.$ This is also denoted by ${}^n{C}_r$ and the objects are unordered.

${\displaystyle \left(\genfrac{}{}{0px}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}.}$