A very small population consists of 7 units, labeled a; b; c; d; e; f; g. How many
di�fferent samples of size 3 are there, for sampling without replacement, assuming
that
(a) samples are lists of units.
(b) samples are sets of units.
To determine the number of different samples of size 3 from a population of 7 units, we can calculate it separately for both scenarios:
(a) Samples as Lists:
When samples are considered as lists, the order of units in the sample matters. This means that "abc" and "bac" would be considered as different samples, even though they contain the same units.
To calculate this, we can use the formula for permutations without repetition. The number of different samples can be determined by calculating the number of permutations of 7 units taken 3 at a time.
We can use the formula for permutations without repetition to calculate this:
nPr = n! / (n - r)!
Where n is the total number of units in the population (7) and r is the size of the sample (3).
Using this formula, we can calculate:
7P3 = 7! / (7 - 3)!
= 7! / 4!
Since '!' represents the factorial function, the formula expands to:
7P3 = 7 * 6 * 5
Calculating this gives us:
7P3 = 210
Therefore, there are 210 different samples of size 3 when samples are considered as lists.
(b) Samples as Sets:
When samples are considered as sets, the order of units in the sample does not matter. This means that "abc" and "bac" would be considered as the same sample, as they contain the same units.
To calculate this, we can use the formula for combinations without repetition. The number of different samples can be determined by calculating the number of combinations of 7 units taken 3 at a time.
We can use the formula for combinations without repetition to calculate this:
nCr = n! / (r! * (n - r)!)
Where n is the total number of units in the population (7) and r is the size of the sample (3).
Using this formula, we can calculate:
7C3 = 7! / (3! * (7 - 3)!)
= 7! / (3! * 4!)
Since '!' represents the factorial function, the formula expands to:
7C3 = 7 * 6 * 5 / (3 * 2 * 1)
Calculating this gives us:
7C3 = 35
Therefore, there are 35 different samples of size 3 when samples are considered as sets.