Question

select the correct answer. an art gallery wants to display 4 pieces of art in the front window. if there are 8 pieces to choose from, how many distinct displays are possible? a. 40,320 b. 10,080 c. 1,680 d. 70

Explanation:

Step1: Identify permutation need

Since the order of art pieces in a display matters (distinct arrangements count as different displays), we use permutations. The formula for permutations of $n$ items taken $k$ at a time is $P(n,k)=\frac{n!}{(n-k)!}$ where $n=8$, $k=4$.

Step2: Calculate factorial values

First, compute $n!=8!=8\times7\times6\times5\times4\times3\times2\times1=40320$
Then compute $(n-k)!=(8-4)!=4!=4\times3\times2\times1=24$

Step3: Compute permutation value

Substitute into the permutation formula:
$P(8,4)=\frac{8!}{(8-4)!}=\frac{40320}{24}=1680$

Answer:

C. 1,680