For a game, friends Arnaldo, Bianca, Carlos, Denise and Elana decided to create a circle in how many different ways can this circle be organized?

To solve the problem of determining how many different ways the circle can be organized, we'll use the concept of circular permutations.

In a circular permutation of \( n \) objects, the number of distinct arrangements is given by:

\[
(n-1)!
\]

This is because, in a circle, one position can be fixed (to avoid counting identical rotations as different arrangements).

Given the problem, there are 5 friends: Arnaldo, Bianca, Carlos, Denise, and Elana.

So, \( n = 5 \).

Thus, the number of distinct arrangements is:

\[
(5-1)! = 4! = 24
\]

So, there are **24 different ways** to organize this circle.

Would you like more details, or do you have any questions?

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Here are 8 related questions:

1. How would the number of arrangements change if there were 6 friends instead?
2. What is the formula for linear permutations, and how does it differ from circular permutations?
3. How many ways can 3 people arrange themselves in a circle?
4. What if the circle is not closed? How many ways can the 5 friends be arranged in a line?
5. How would adding constraints (e.g., certain people must sit next to each other) change the number of arrangements?
6. How would the result change if one person is fixed in a specific position?
7. What is the difference between a combination and a permutation in this context?
8. How do the concepts of symmetry and rotation affect the calculation of circular permutations?

**Tip:** Remember that in circular permutations, the position of the first object is usually fixed to avoid counting identical rotations multiple times.

Learn how to calculate circular permutations with friends Arnaldo, Bianca, Carlos, Denise, and Elana. Discover the number of ways to organize them in a circle using the factorial formula (n-1)!. Suitable for Grades 9-12.

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