Question

The number of possible samples of size n out of N population units in simple random sampling without replacement is

• A. $N^n$
• B. $^N C_n$
• C. $n!$
• D. $n^N$

Hint:

Associate "Without Replacement" (SRSWOR) with Combinations ($^N C_n$) and "With Replacement" (SRSWR) with power products ($N^n$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In Simple Random Sampling Without Replacement (SRSWOR), a subset of $n$ unique units is selected from a population of $N$ units. The order of selection does not matter.

Step 2: Detailed Explanation:
Because selection is performed without replacement, no unit can be chosen more than once.
Since the order of units in the final sample is irrelevant, we use the mathematical combination formula to find the number of unique sample subsets.
The number of ways to choose $n$ items from a pool of $N$ items is given by:
\[ \binom{N}{n} = ^N C_n = \frac{N!}{n!(N-n)!} \]
If sampling were performed with replacement (SRSWR), the number of possible samples would be $N^n$.
Therefore, the number of possible samples under SRSWOR is $^N C_n$.

Step 3: Final Answer
The correct option is (B).