Permutasi siklis

Summary

This video tutorial covers the concept of cyclic permutations and contrasts them with regular permutations. It explains how to calculate the number of seating arrangements in various scenarios, including with restrictions such as people needing to sit together. Key topics include how cyclic permutations differ by subtracting one from the total number of items and how to apply factorial calculations in different contexts. Examples include seating arrangements for families, groups of friends, and work teams, with some involving specific seating requirements like individuals always sitting next to each other or alternating between genders.

Takeaways

• 😀 Regular permutations are used for arranging objects in a line, calculated using n! (factorial).
• 😀 Cyclic permutations are used for arranging objects in a circle, with the formula (n-1)!.
• 😀 The key difference between regular and cyclic permutations is the treatment of rotational symmetry in circles.
• 😀 When constraints are introduced, such as people needing to sit next to each other, they are treated as a single unit in permutation calculations.
• 😀 In cyclic permutations, if some objects are fixed in place (such as the head of a table), the calculation adjusts accordingly.
• 😀 For problems with specific seating arrangements (like parents with children), the constraints reduce the total number of groups or units to be arranged.
• 😀 When people must sit next to specific others, their seating arrangements are calculated as a combined unit, reducing the problem's complexity.
• 😀 The cyclic permutation formula is especially useful when the objects being arranged are part of a circle, as the position of one object can be fixed to simplify calculations.
• 😀 Permutations with multiple constraints (e.g., several people or groups must sit together) involve multiplying factorials to account for different arrangements.
• 😀 Advanced scenarios, like alternating men and women in seating arrangements, require locking one group and using cyclic permutations for others.
• 😀 Special cases, such as when a position (e.g., a chairman) is fixed, require adjusting the number of units for permutation, leading to a reduced factorial calculation.

Q & A

What is the difference between regular permutations and cyclic permutations?

In regular permutations, the objects are arranged in a linear sequence, and the number of ways to arrange them is calculated by n!. In cyclic permutations, the objects are arranged in a circle, and the number of ways to arrange them is reduced by fixing one position, so the formula becomes (n-1)!.

How do you calculate the number of seating arrangements for 4 people sitting in a row?

The number of ways to arrange 4 people in a row is calculated by 4!, which equals 4 × 3 × 2 × 1 = 24.

How is the number of seating arrangements different when 4 people are sitting in a circle?

In a circle, the number of arrangements is reduced because rotations of the same arrangement are considered equivalent. For 4 people, the number of arrangements is (4-1)! = 3!, which equals 3 × 2 × 1 = 6.

Why do we subtract 1 from n in cyclic permutations?

In cyclic permutations, subtracting 1 accounts for the fact that rotating the arrangement does not result in a new unique configuration. Fixing one position eliminates identical rotations.

How would you approach a permutation problem where there are restrictions, such as a mother and child who must sit together?

Treat the mother and child as a single block or unit. Then, calculate the number of ways to arrange the remaining objects and multiply by the number of ways the mother and child can be arranged within their block.

If 5 people need to sit in a circle with the mother and child always together, how many seating arrangements are possible?

If the mother and child are treated as a single unit, there are 4 units to arrange in a circle, so the number of arrangements is (4-1)! = 3! = 6. Additionally, the mother and child can be arranged in 2! = 2 ways within their unit. Therefore, the total number of arrangements is 6 × 2 = 12.

What is the number of arrangements when three friends must sit together in a circle among 7 people?

Treat the three friends as one unit, leaving 5 units to arrange in a circle. The number of ways to arrange the 5 units is (5-1)! = 4!, which equals 24. The three friends can be arranged among themselves in 3! = 6 ways. Therefore, the total number of arrangements is 24 × 6 = 144.

In a situation where 4 married couples must sit together in a circle, how do you calculate the number of seating arrangements?

Treat each married couple as a unit, reducing the problem to arranging 4 units in a circle. The number of arrangements is (4-1)! = 3!. Since each couple can be arranged in 2! ways within their unit, the total number of arrangements is 3! × 2! × 2! × 2! × 2! = 6 × 16 = 96.

How do you handle seating arrangements when two groups (men and women) must sit together but separately?

Treat each group as a single block or unit. In this case, with two blocks to arrange, the number of arrangements is (2-1)! = 1!. Then, arrange the individuals within each group, calculating the arrangements separately for the men and women.

What is the number of ways to arrange 3 men and 3 women in a circle if the women must sit together?

Treat the women as one block, reducing the problem to arranging 4 units in a circle. The number of arrangements is (4-1)! = 3!, which equals 6. The women can be arranged among themselves in 3! = 6 ways. Therefore, the total number of arrangements is 6 × 6 = 36.