select the correct answer. an art gallery wants to display 4 pieces of art in the front window. if there are…

Explanation:

Step1: Identify the permutation formula

The problem is about permutations. The formula for permutations is $P(n,r)=\frac{n!}{(n - r)!}$, where $n$ is the total number of items and $r$ is the number of items to be selected. Here, $n = 8$ and $r=4$.

Step2: Calculate factorial values

$n!=8! = 8\times7\times6\times5\times4\times3\times2\times1$, $(n - r)!=(8 - 4)!=4!=4\times3\times2\times1$. Then $P(8,4)=\frac{8!}{(8 - 4)!}=\frac{8!}{4!}=\frac{8\times7\times6\times5\times4!}{4!}$.

Step3: Simplify the expression

Cancel out the $4!$ terms. We get $P(8,4)=8\times7\times6\times5=1680$.

Answer:

C. 1,680