Question

The Cleveland Art Institute has eight paintings by a local artist they plan to display. If the paintings are hung in one horizontal line, how many different ways can they be arranged if order is important?

40,320

20,160

10,880

1

Answer 1

To determine the number of different ways to arrange 8 paintings in a horizontal line, we can use the concept of permutations. The number of permutations of \( n \) items is calculated using the factorial function, denoted as \( n! \).

For \( n = 8 \) paintings, the number of arrangements (permutations) is calculated as follows:

\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\]

Let's compute this step by step:

1. \( 8 \times 7 = 56 \)
2. \( 56 \times 6 = 336 \)
3. \( 336 \times 5 = 1680 \)
4. \( 1680 \times 4 = 6720 \)
5. \( 6720 \times 3 = 20160 \)
6. \( 20160 \times 2 = 40320 \)
7. \( 40320 \times 1 = 40320 \)

Thus, \( 8! = 40320 \).

The correct answer is:

\[
\boxed{40320}
\]