We're going to be using combination since this question is asking how many different combinations of 10 people can be selected from a set of 23.
We would only use permutation if the order of the people in the committee mattered, which it seems it doesn't.
Formula for combination:
[tex]C(n,r)=\dfrac{n!}{(n-r)!r!}[/tex]
Where [tex]n[/tex] represents the number of objects/people in the set and [tex]r[/tex] represents the number of objects/people being chosen from the set
There are 23 people in the set and 10 people being chosen from the set
[tex]C(23,10)=\dfrac{23!}{(23-10)!10!}[/tex]
[tex]=\dfrac{23!}{13!\times10!}[/tex]
Usually I would prefer solving such fractions by hand instead of a calculator, but factorials can result in large numbers and there is too much multiplication. Using a calculator, we get
[tex]=1,144,066[/tex]
Thus, there are 1,144,066 different 10 person committees that can be selected from a pool of 23 people. Let me know if you need any clarifications, thanks!
~ Padoru
