Permutación Circular - Ejercicios Resueltos

Summary

In this educational video, the instructor explains circular permutations through engaging examples, starting with the arrangement of 6 friends around a circular table. The video explores handling restrictions, such as when a couple must sit together, and calculates the number of ways to assign colors to a circular logo using permutation formulas. Emphasizing the use of addition and multiplication principles in combinatorics, the instructor guides viewers through clear, practical applications of circular permutations and combinations, making complex concepts more accessible for exam preparation.

Takeaways

• 😀 The script introduces circular permutations, which are used to arrange elements in a circle, such as seating friends around a campfire or at a circular table.
• 😀 The formula for calculating circular permutations of n elements is (n-1)!, where n is the total number of elements to be arranged.
• 😀 A fixed reference point (such as a specific element) is essential in circular permutations because the circle has no inherent start or end point.
• 😀 In circular permutations, the order of elements matters, and all elements must participate in the arrangement.
• 😀 The first problem in the video calculates the number of ways six friends can sit around a circular table using the circular permutation formula (n-1)!, resulting in 120 arrangements.
• 😀 The second problem introduces a scenario where a pair of friends (a couple) must always sit together. This scenario involves treating the couple as a single block and then calculating the number of ways to arrange five elements (four friends and the couple as one block).
• 😀 The principle of multiplication is applied in permutation problems where multiple steps are involved, such as first arranging the five elements and then considering the two possible arrangements of the couple.
• 😀 In the second problem, once the couple is treated as a block, we calculate the circular permutation of the five blocks, resulting in 24 possible arrangements. Then, we multiply by 2 to account for the two seating arrangements within the couple block, leading to 48 total arrangements.
• 😀 The third problem demonstrates how to calculate the number of ways to paint a logo with seven different colors. There are seven positions (six around a center), and each circle must be painted a different color.
• 😀 The principle of multiplication is again used in the logo painting problem. First, the center circle is painted, and then the six surrounding circles are arranged in a circular manner, leading to 840 total ways to paint the logo.
• 😀 The video emphasizes the importance of understanding permutation and combination principles, as well as applying them in various problem scenarios to solve complex arrangements and counting problems.

Q & A

What is the formula for calculating circular permutations?

The formula for circular permutations is (n-1)!, where n is the number of elements being arranged in a circle.

Why do we subtract 1 from the total number of elements in a circular permutation?

We subtract 1 from the total number of elements because in circular permutations, one element must be fixed to avoid counting rotationally identical arrangements as distinct.

How do circular permutations differ from linear permutations?

Circular permutations account for the fact that the arrangement is in a circle, meaning rotations of the same arrangement are not considered different. In linear permutations, the order of elements matters, and rotations of the same arrangement are considered different.

How is the problem of arranging 6 friends around a circular table solved?

For 6 friends, the number of ways they can be arranged around a circular table is given by (6-1)!, which equals 5! = 120 ways.

What strategy is used to solve the problem of Carlos and Sandra sitting together at a circular table with four other friends?

Carlos and Sandra are treated as a single 'block,' which reduces the problem to arranging 5 elements (the block and 4 friends). The number of ways to arrange them is 4!, and since Carlos and Sandra can switch places within their block, we multiply by 2, giving a total of 48 arrangements.

Why do we treat Carlos and Sandra as a 'block' when solving the problem?

Treating Carlos and Sandra as a 'block' simplifies the problem by reducing the number of elements to be arranged. Since they must sit together, this method allows us to calculate their arrangement as a single unit.

How do we calculate the number of ways to color the logo with 7 colors?

To color the logo with 7 colors, we first choose a color for the central circle. For the remaining 6 circles, we calculate the number of ways to arrange 6 colors in a circle, which is 6! = 720. Combining these steps, the total number of color arrangements is 7 * 720 = 840.

What is the importance of the multiplication principle in solving the logo coloring problem?

The multiplication principle is used to combine the two independent events in the logo coloring problem: first choosing a color for the central circle and then arranging the remaining 6 colors in a circular manner. These two events are multiplied to get the total number of arrangements.

How do we handle the rotational symmetry in circular arrangements when calculating permutations?

In circular arrangements, one element is fixed to break the symmetry and avoid counting identical rotations as distinct. This reduces the total number of arrangements compared to linear permutations.

What is the significance of learning about circular permutations for students preparing for exams?

Learning about circular permutations helps students understand a key concept in combinatorics, which is often tested in exams. It provides insight into how rotational symmetry affects the number of possible arrangements, making it a crucial concept for problem-solving in various scenarios.