Question

How many samples of size n=3 can be selected from a population with the following sizes: N=4, N=8, N=20, N=50? A population consists of the five numbers 2, 3, 6, 8 and 11. Consider samples of size 2 that can be drawn from this population.

Answer 1

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

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

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

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

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

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

$\begin{pmatrix} 5\\ 2 \end{pmatrix}=\cfrac{5! } {2! \cdot3! }=\cfrac{4\cdot5}{2}=10.$

The possible samples:

(2, 3), (2, 6), (2, 8), (2, 11), (3, 6),

(3, 8), (3, 11), (6, 8), (6, 11), (8, 11).