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

Explanation:

Step1: Identify the problem type

This is a permutation problem since the order of the art pieces in the display matters (distinct displays imply different orderings). The formula for permutations of ( n ) objects taken ( r ) at a time is ( P(n, r)=\frac{n!}{(n - r)!} ). Here, ( n = 8 ) (total art pieces) and ( r = 4 ) (pieces to display).

Step2: Calculate the permutation

First, find ( 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\times7\times6\times5\times4!}{4!} ). The ( 4! ) terms cancel out, so we have ( 8\times7\times6\times5 ).

Calculate ( 8\times7 = 56 ), ( 56\times6 = 336 ), ( 336\times5 = 1680 )? Wait, no, wait. Wait, no, permutation formula: Wait, no, wait, ( 8\times7\times6\times5 = 1680 )? Wait, no, 87=56, 566=336, 3365=1680? But wait, let's recalculate: 8P4 is 8!/(8-4)! = 8!/4! = (8×7×6×5×4!)/4! = 8×7×6×5 = 1680? But wait, the options: option C is 1680? Wait, but wait, maybe I made a mistake. Wait, no, wait, 8P4: 8765 = 1680. But let's check the options. Option C is 1680. Wait, but let's confirm.

Wait, no, wait, maybe it's a permutation. Wait, the problem says "distinct displays", so order matters. So permutation. So 8P4 = 8!/(8-4)! = 876*5 = 1680. So the answer should be C.

Wait, but wait, let's check again. 87=56, 566=336, 336*5=1680. Yes. So the correct answer is C.

Answer:

C. 1,680