How many different simple random samples of size $3$ can be selected from a population of size $7$?

• A. $14$
• B. $42$
• C. $21$
• D. $35$

Answer: (D)

Verified step by step guidance

1. Understand that the problem asks for the number of different simple random samples of size 3 from a population of size 7. This means we want to find the number of combinations of 7 items taken 3 at a time.
2. Recall the formula for combinations (also called "n choose r"), which is used when the order of selection does not matter: \[{{n}\choose{r}} = \frac{n!}{r! (n-r)!}\] where \(n\) is the population size and \(r\) is the sample size.
3. Substitute the given values into the formula: \[{{7}\choose{3}} = \frac{7!}{3! (7-3)!} = \frac{7!}{3! 4!}\]
4. Calculate the factorial expressions step-by-step, but do not compute the final number: - \(7! = 7 \times 6 \times 5 \times 4!\) - \(3! = 3 \times 2 \times 1\) - \$4!$ remains as is to cancel out with the numerator.
5. Simplify the expression by canceling \$4!$ in numerator and denominator, then multiply and divide the remaining numbers to find the total number of different samples.