Answer to Question #338223 in Statistics and Probability for Flora

Question

How many different samples of size 8 can be selected from a population with a size of 12 ?

Answer 1

Consider two cases: samples without replacement (the order of elements in samples does not matter) and samples with replacement (the order of elements in the sample matters). In the first case we receive $C_{12}^8=\frac{12!}{8!4!}=\frac{9\cdot10\cdot11\cdot12}{4!}={9\cdot5\cdot11}=495$ . The number of different samples is equal to the number of subsets of size $8$ of the set of size $12$. The number of samples with replacement (the order of elements matters) is equal to $12^8$. We used the multiplication principle of combinatorics.

Answer: there are $495$ samples without replacement(the order of elements does not matter) and $12^8$ samples with replacement(the order of elements matters).