(peluang) permutasi siklis.flv

Summary

This video explains the concept of cyclic permutations, focusing on scenarios where people or objects are arranged in a circular fashion. It uses examples like people sitting in a circle or arranging objects in a bracelet-like pattern. The presenter introduces the formula for calculating cyclic permutations (N-1 factorial) and applies it to problems involving different numbers of people, demonstrating how to determine the number of unique seating arrangements. The script also covers more complex situations, such as when specific people must sit together, leading to a step-by-step breakdown of the solution.

Takeaways

• 😀 Circular permutation involves arrangements where the elements are seated in a circle, such as people sitting around a table or items arranged in a circular fashion.
• 😀 The formula for calculating the number of arrangements in circular permutation is (N-1)! where N is the total number of elements (people, items, etc.).
• 😀 Example: With 4 people sitting in a circle, the number of arrangements is (4-1)! = 3! = 6.
• 😀 If the arrangement is 'free', where there is no restriction on seating, the number of ways to arrange N elements is (N-1)!, calculated using the factorial function.
• 😀 When there are 3 men and 3 women to be seated in a circle, the number of ways to arrange them without any restriction is (6-1)! = 5! = 120.
• 😀 If there is a restriction where 3 men must sit together, they are considered as a single block, reducing the problem to arranging 4 blocks (3 men as one and 3 women). The number of arrangements then becomes (4-1)! = 3! = 6.
• 😀 Once the men are treated as a block, the arrangements of the men within their block also need to be considered, and the number of ways to arrange the men within this block is 3! = 6.
• 😀 Therefore, the total number of seating arrangements where 3 men must sit together is the product of (3! for the block arrangement) and (3! for the men within the block), yielding 6 * 6 = 36.
• 😀 Circular permutation reduces the number of arrangements compared to linear permutation because of the rotational symmetry of a circle (i.e., rotating the entire arrangement does not create a new unique arrangement).
• 😀 Understanding circular permutation is useful in real-life applications such as seating arrangements, scheduling, and designing circular structures.

Q & A

What is the main topic discussed in the video?

The main topic discussed is cyclic permutations, specifically how arrangements of people or objects in a circle can be calculated.

What is a cyclic permutation?

A cyclic permutation refers to an arrangement of elements in a circle, where the arrangement can be rotated without creating a new distinct permutation.

What is the formula for calculating the number of distinct cyclic permutations?

The formula for calculating the number of distinct cyclic permutations is (n - 1)!, where 'n' is the number of elements or people in the circle.

Why is the formula (n - 1)! used instead of n!?

The formula (n - 1)! is used because in cyclic arrangements, rotating the arrangement does not create a new distinct permutation, hence reducing the total number of permutations by a factor of n.

If 4 people are seated in a circle, how many distinct seating arrangements are possible?

For 4 people, the number of distinct seating arrangements is (4 - 1)! = 3! = 6.

How would you calculate the number of ways to arrange 3 men and 3 women in a circle with no restrictions?

For 6 people (3 men and 3 women), the number of distinct seating arrangements is (6 - 1)! = 5! = 120.

How does the seating arrangement change when the 3 men must sit together?

If the 3 men must sit together, they are treated as a single unit. So, instead of 6 units, we have 4 units to arrange, leading to (4 - 1)! = 3! = 6 distinct arrangements for the units. Additionally, the 3 men can be arranged among themselves in 3! = 6 ways, so the total number of arrangements is 6 * 6 = 36.

Why are the 3 men treated as a single unit in the arrangement?

The 3 men are treated as a single unit because the requirement is for them to sit together, which means they must be arranged as one block or group within the circle.

In the case of the 3 men sitting together, how many possible seating arrangements exist for the women?

For the women, there are 3! = 6 distinct ways to arrange them, as there are no restrictions on their seating other than that they are in a circle.

What is the total number of seating arrangements when the 3 men must sit together and the 3 women have no restrictions?

The total number of seating arrangements is 36, which is calculated by multiplying the distinct arrangements of the 4 units (6) by the distinct arrangements of the 3 men (6), resulting in 6 * 6 = 36.