Arrangements of n different things taken r at a time:

To solve this problem, we need to understand the concept of permutations. Permutations refer to the arrangement of objects in a specific order. In this case, we want to find the number of arrangements of n different things taken r at a time.

Answer Explanation:

The correct answer is option A, which states that the number of arrangements is P(n-1, r).

Explanation:

1. Permutations formula:
The number of permutations of n objects taken r at a time is given by the formula:
P(n, r) = n! / (n-r)!
This formula calculates the number of ways to arrange n objects in a specific order, taking only r objects at a time.

2. Excluding a particular thing:
In the given problem, we want to find the number of arrangements that exclude a particular thing. This means that we have n-1 objects to arrange instead of n objects.

3. Using the permutations formula:
By substituting n-1 for n in the permutations formula, we get:
P(n-1, r) = (n-1)! / ((n-1)-r)!
Simplifying further, we have:
P(n-1, r) = (n-1)! / (n-r-1)!

This formula calculates the number of arrangements of n-1 objects taken r at a time, which is the same as excluding a particular thing from the original set of n objects.

4. Conclusion:
Therefore, the number of arrangements of n different things taken r at a time, excluding a particular thing, is given by the formula P(n-1, r), which is option A.

Note:
The other options, P(n, r) and nP(n-1, r-1), do not accurately represent the number of arrangements in this scenario. P(n, r) calculates the number of arrangements without excluding any particular thing, while nP(n-1, r-1) represents the number of permutations of n-1 objects taken r-1 at a time, which is not the same as excluding a particular thing.