ⁿPᵣ equals:

A permutation is an ordered arrangement of a set of objects. The notation $^nP_r$ (or $P(n, r)$) represents the number of ways to choose and arrange $r$ distinct items from a set of $n$ distinct items where the order of selection matters. ### Step-by-Step Derivation: 1. **First Position:** There are $n$ possible choices. 2. **Second Position:** Since one item is already used, there are $(n-1)$ choices. 3. **$r$-th Position:** Following the pattern, the $r$-th position has $(n - r + 1)$ choices. ### Mathematical Expression: By the Fundamental Counting Principle, the total number of permutations is: $$^nP_r = n \times (n - 1) \times (n - 2) \times \dots \times (n - r + 1)$$ To simplify this into factorial notation, we multiply and divide by $(n - r)!$: $$^nP_r = \frac{n \times (n - 1) \times \dots \times (n - r + 1) \times (n - r) \times (n - r - 1) \times \dots \times 1}{(n - r) \times (n - r - 1) \times \dots \times 1}$$ This simplifies to: $$^nP_r = \frac{n!}{(n - r)!}$$ Comparing this with the given options, the correct choice is **C**.