High School

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Small towns frequently erect signs in the fields alongside the highway to encourage passersby to stop and shop. If each of 8 businesses place one sign in a line along the side of the highway, in how many different orders might the signs be arranged?

Answer :

Final Answer:

There are 40,320 different orders in which the signs can be arranged.

Explanation:

To calculate the number of different orders in which the 8 signs can be arranged, you can use the concept of permutations. In this case, you have 8 businesses, and you want to arrange their signs in a line alongside the highway. The number of different orders is given by 8 factorial, denoted as 8!.

8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

So, there are 40,320 different orders in which the signs can be arranged along the side of the highway, each representing a unique arrangement of businesses' signs to encourage passersby to stop and shop.

Permutations are a fundamental concept in combinatorics, representing the different ways you can arrange a set of objects. The factorial notation, such as 8!, is commonly used to express permutations. It's essential for solving problems involving order and arrangement of elements.

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