Question

How many different samples of size n = 3 can be selected from a population with the following sizes?

1.) N = 4

2.) N = 8

3.) N = 20

4.) N = 50

Answer 1

The number of possible samples which can be selected without replacement is

$\begin{pmatrix} N \\ n \end{pmatrix}=\cfrac{N! } {n! \cdot(N-n)! }.$

$1.) \ \begin{pmatrix} 4 \\ 3 \end{pmatrix}=\cfrac{4! } {3! \cdot1! }=4.$

$2.) \ \begin{pmatrix} 8\\ 3 \end{pmatrix}=\cfrac{8! } {3! \cdot5! }=\\ =\cfrac{6\cdot7\cdot8}{2\cdot3}=56.$

$3.) \ \begin{pmatrix} 20\\ 3 \end{pmatrix}=\cfrac{20! } {3! \cdot17! }=\\ =\cfrac{18\cdot19\cdot20}{2\cdot3}=1140.$

$4.) \ \begin{pmatrix} 50\\ 3 \end{pmatrix}=\cfrac{50! } {3! \cdot47! }=\\ =\cfrac{48\cdot49\cdot50}{2\cdot3}=19600.$