Let's break down the two parts of the problem:

1. With repetition allowed:

A 4-digit number is formed by selecting digits from the set $\{0, 1, 2, 3, 4, 5\}$, where repetition is allowed.

• The first digit of the number cannot be $0$ (since it must be a 4-digit number), so we have 5 choices: $\{1, 2, 3, 4, 5\}$.
• The second, third, and fourth digits can be any of the 6 digits, including $0$.

Thus, the total number of 4-digit numbers that can be formed is: $5 \times 6 \times 6 \times 6 = 5 \times 6^3 = 5 \times 216 = 1080.$

2. Without repetition allowed:

When repetition is not allowed, we need to choose distinct digits for each position.

• The first digit can still be any of $\{1, 2, 3, 4, 5\}$, giving us 5 choices.
• The second digit can be any digit except the first digit, so we have 5 choices (since 0 is allowed).
• The third digit can be any of the remaining 4 digits.
• The fourth digit can be any of the remaining 3 digits.

Thus, the total number of 4-digit numbers that can be formed without repetition is: $5 \times 5 \times 4 \times 3 = 5 \times 5 \times 12 = 300.$

Conclusion:

• With repetition: 1080
• Without repetition: 300

So, the correct answer is b. 1080, 300.

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Would you like any further explanation or clarification on any of the steps? Here are some related questions:

1. How would the number of possible 4-digit numbers change if the digit set was $\{0, 1, 2, 3, 4\}$?
2. What if the number of digits allowed was reduced to 3 instead of 4?
3. How would the answer change if we allowed negative numbers?
4. What if the set of digits was extended to $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$?
5. Can you explain why the first digit cannot be 0 in a 4-digit number?

Tip: Always remember that when repetition is allowed, each position can be filled independently of the others, but when repetition is not allowed, each digit must be distinct from the others.