Kaidah Pencacahan 4 - Permutasi Unsur yang Sama dan Permutasi Siklis Matematika Wajib Kelas 12

Summary

In this video, Deni Handayani explains the concepts of 'permutasi unsur yang sama' (permutations of identical elements) and 'permutasi siklis' (circular permutations) as part of a series on kaidah pencacahan. He covers how to calculate permutations when some elements are identical, using examples like arranging the letters of 'Belibis' and books of different subjects. The video also explores circular permutations, with examples including seating arrangements for groups at a round table. Through clear explanations and practical examples, Deni provides a comprehensive understanding of these topics in combinatorics.

Takeaways

• 😀 Takeaway 1: The video covers the fourth part of counting principles, focusing on permutation, repeated elements, and cyclic permutations.
• 😀 Takeaway 2: The presenter explains the formula for permutations with repeated elements: P = n! / (n1! * n2! * ... * nk!), where n represents the total number of elements and n1, n2, etc., are the frequencies of identical elements.
• 😀 Takeaway 3: An example is given with the word 'Belibis,' where there are repeated letters (B and E). The number of distinct permutations is calculated using the formula for repeated elements, resulting in 1260 distinct arrangements.
• 😀 Takeaway 4: In the second example, eight books (4 math, 2 physics, and 2 biology) are arranged, and the total number of ways to arrange them is found to be 420 using the same permutation formula for repeated elements.
• 😀 Takeaway 5: The video introduces cyclic permutations, where elements are arranged in a circular manner. The formula for cyclic permutations is P = (n-1)!, where n is the total number of elements.
• 😀 Takeaway 6: In an example involving six people at a circular table, the number of distinct seating arrangements is calculated using the cyclic permutation formula, resulting in 120 different seating positions.
• 😀 Takeaway 7: Another cyclic permutation example involves seven distinct-colored beads arranged on a bracelet. The number of ways to arrange these beads is found to be 720 using the cyclic permutation formula.
• 😀 Takeaway 8: The video discusses a scenario where a father, mother, and three children sit at a circular table, with the condition that the father and mother must sit next to each other. The total number of arrangements is 12, considering both cyclic permutations and the swapping of the father and mother.
• 😀 Takeaway 9: In a similar scenario with six people sitting at a circular table, three people must sit together. This is treated as one block, and the number of distinct seating arrangements is calculated to be 36, considering both cyclic permutations and internal swapping within the block.
• 😀 Takeaway 10: The presenter encourages viewers to review the previous videos for a better understanding of permutations and combinations, which form the basis for the discussed concepts.

Q & A

What is the formula for permutations with identical elements?

The formula for permutations with identical elements is: P = n! / (n1! * n2! * ... * nk!), where n is the total number of elements and n1, n2, ..., nk are the frequencies of each repeated element.

How do you calculate the number of distinct arrangements for the word 'Belibis'?

To calculate the number of distinct arrangements for 'Belibis', we use the formula for permutations with identical elements: P = 7! / (2! * 2!). This results in 1260 distinct arrangements.

In the book arrangement example, why is it necessary to consider identical books?

In the book arrangement example, identical books must be considered to avoid overcounting arrangements where books of the same type are swapped, which would result in indistinguishable outcomes.

What is the difference between a regular permutation and a cyclic permutation?

A regular permutation involves arranging elements in a straight line, while a cyclic permutation involves arranging elements in a circle, where rotating the arrangement does not produce a new unique configuration.

What formula is used to calculate cyclic permutations?

The formula for cyclic permutations is P = (n - 1)!, where n is the total number of elements to be arranged in a circle.

How many distinct seating arrangements can be made for 6 people around a round table?

For 6 people seated around a round table, the number of distinct seating arrangements is calculated using cyclic permutations: P = (6 - 1)! = 5! = 120.

How do we calculate the number of arrangements when people must sit next to each other in cyclic permutations?

When certain people must sit next to each other in cyclic permutations, we treat them as a single unit. This reduces the number of elements to arrange, and we then calculate the cyclic permutation for the reduced number of units.

How many distinct ways can 7 different colored beads be arranged on a bracelet?

For 7 different colored beads on a bracelet (a circular arrangement), the number of distinct arrangements is calculated using the cyclic permutation formula: P = (7 - 1)! = 6! = 720.

What happens when three people must always sit next to each other in a cyclic arrangement of 6 people?

When three people must always sit next to each other, we treat them as a single unit, reducing the number of units to 4. Then, the cyclic permutation formula is applied: P = (4 - 1)! = 3! = 6. We multiply by 3! to account for the internal arrangement of the three people, resulting in 36 distinct seating arrangements.

In the case where a father and mother must sit together, how does this affect the permutation calculation?

When the father and mother must sit together, they are treated as one combined unit, reducing the number of elements to arrange. The number of distinct seating arrangements is then calculated by applying the cyclic permutation formula and considering the internal arrangements of the father and mother.