Which of the following statements are true? Justify your answers. This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. The total number of all possible samples of size 3 without replacement from a population of size 7 is 21 .

To evaluate the statement “The total number of all possible samples of size 3 without replacement from a population of size 7 is 21,” we need to use the concept of combinations.

Combination Formula

The number of ways to choose a sample of size \(r\) from a population of size \(n\) without replacement is given by the combination formula:

\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

where \(n!\) denotes the factorial of \(n\).

Application to the Statement

In this case, we have a population size \(n = 7\) and a sample size \(r = 3\). Plugging these values into the combination formula:

\[
C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \times 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

Conclusion

The statement “The total number of all possible samples of size 3 without replacement from a population of size 7 is 21” is false. The correct number of samples is 35.