select the correct answer. an art gallery wan...
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
Answer
# 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