Step 1: Determine the number of possibilities for the first digit.

The digits are selected from the set $\{1, 2, 3, 4, 5\}$. Since we are selecting one digit and there are 5 digits in total, there are 5 possibilities for the first digit.

Step 2: Determine the number of possibilities for the second digit.

The selection is made with replacement, meaning the second digit is also chosen from the same set $\{1, 2, 3, 4, 5\}$. Hence, there are again 5 possibilities for the second digit.

Step 3: Use the counting principle to find the total number of elements in the sample space.

The counting principle states that if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Here:

• There are 5 possibilities for the first digit.
• There are 5 possibilities for the second digit.

Thus, the total number of elements in the sample space is: $5 \times 5 = 25$

Final Answer:

There are 25 possible outcomes in the sample space when selecting two digits with replacement from the set $\{1, 2, 3, 4, 5\}$.

Would you like further details on this, or do you have any questions?

Here are 5 related questions:

1. What if the selection was without replacement?
2. How many outcomes would there be if three digits were selected instead of two?
3. What is the probability of selecting two identical digits?
4. How would the sample space change if the set included digits 0–9?
5. How many outcomes are there if the digits must be distinct?

Tip: The counting principle can be applied to various multi-step problems to simplify the process of calculating total outcomes.