The question appears to be about permutations where repetition is allowed. The problem asks for the number of ways to arrange $r$ objects out of $n$ objects, given that repetitions are allowed.

For this type of situation, the formula used to calculate the number of permutations when repetition is allowed is:

$n^r$

This means there are $n$ choices for each of the $r$ positions, leading to $n \times n \times \dots \times n$ (a total of $r$ times), which simplifies to $n^r$.

Now let's examine the options:

• (a) $n \cdot {}_C^r$: This is not correct as this represents a combination and a product, not dealing with repetition in permutations.
• (b) $n^r = {}_C^r + 1$: This is incorrect because $n^r$ does not include a combination plus 1.
• (c) $n = {}_P^r \cdot r!$: This is not correct as well, since it appears to confuse the formula for permutations without repetition.
• (d) None of these: Based on the options, it seems this is the correct answer since none of the given formulas match the correct one, $n^r$.

The correct answer is therefore (d) None of these.

Would you like more details on permutations with repetition or related concepts?

Here are five relative questions for expanding your understanding:

1. How do permutations without repetition differ from permutations with repetition?
2. What is the formula for combinations when repetition is allowed?
3. Can you apply permutations with repetition to real-life problems, like passwords or codes?
4. How would the formula change if the problem asked for combinations instead of permutations?
5. What are the key distinctions between permutations and combinations in probability?

Tip: When dealing with arrangements, always check if repetition is allowed or not, as it significantly impacts the formula used.