The problem asks to use the counting principle to determine the number of elements in the sample space when two digits are selected without replacement from the digits 1, 2, 3, and 4.

Solution:

The counting principle states that if one event can occur in m ways and another event can occur in n ways, then the total number of ways both events can occur is m × n.

Here:

• The first digit can be chosen in 4 ways (from the digits 1, 2, 3, 4).
• After the first digit is chosen, the second digit can only be chosen from the remaining 3 digits (since the selection is without replacement).

Therefore, the number of elements in the sample space is: $4 \times 3 = 12$

Thus, there are 12 possible outcomes when selecting two digits without replacement.

Do you have any questions or would you like more details?

Here are 5 related questions for further practice:

1. What would be the sample space if you were selecting three digits without replacement from the digits 1, 2, 3, and 4?
2. How would the problem change if the selection was with replacement?
3. What if the digits were selected in such a way that the order does not matter (combinations instead of permutations)?
4. How would you calculate the probability of selecting the digits 1 and 2 in that order?
5. How many ways can you select 2 digits from 5 digits without replacement?

Tip: When solving problems involving "without replacement," always remember that the number of available choices decreases after each selection.