Explanation:

The number of arrangements of n different things taken r at a time can be calculated using the concept of permutations. In permutations, the order of arrangement matters.

Let's consider the given options and analyze them one by one:

a) P(n-1, r-1):
This option suggests using the permutation formula. However, the formula for permutations is P(n, r) = n! / (n-r)!. In this option, the formula used is P(n-1, r-1), which is not the correct formula for the number of arrangements.

b) nP(n-1, r):
This option suggests using the permutation formula, but it is not the correct formula for the number of arrangements. Additionally, the term "nP(n-1, r)" is not a standard notation for permutations.

c) rP(n-1, r-1):
This option suggests using the permutation formula P(n, r) = n! / (n-r)! to calculate the number of arrangements. Let's understand why this is the correct formula:

- Consider a particular thing that must be included in the arrangement. Since it is fixed, we can ignore it temporarily and focus on the remaining (n-1) things to arrange.
- Now, we need to select r-1 more things from the remaining (n-1) things to complete the arrangement.
- The number of ways to select r-1 things from (n-1) things is given by the combination formula C(n-1, r-1).
- Once the r-1 things are selected, the remaining rth thing (which is fixed) can be placed in r different positions within the arrangement.
- Therefore, the total number of arrangements is given by r * C(n-1, r-1).

So, the correct formula for the number of arrangements of n different things taken r at a time which includes a particular thing is rP(n-1, r-1).

d) none of these:
This option is incorrect because option c) is the correct answer.

Conclusion:
The correct answer is option c) rP(n-1, r-1), which represents the formula for the number of arrangements of n different things taken r at a time which include a particular thing.