PERMUTASI - Jenis Permutasi - Matematika SMA/SMK

Summary

In this video, the presenter explains the concept of permutations, covering various types such as permutations with all elements, partial elements, identical elements, and cyclic permutations. The video provides clear examples, such as calculating the number of different arrangements of letters in the word 'Madiun' and how to determine seating arrangements for five people. The explanation includes the use of formulas like factorials and how they relate to order and selection, with practical applications like forming committees and creating distinct number combinations. The presenter also briefly mentions the difference between permutations and combinations.

Takeaways

• 😀 Permutation is a method of counting different arrangements of elements, which can be categorized in various types.
• 😀 Permutation with all elements is calculated using the formula n! (n factorial), where 'n' is the total number of elements.
• 😀 A practical example of permutation with all elements is forming different arrangements from the word 'Madiun'. There are 6 distinct letters, so the total permutations are 6! = 720.
• 😀 For situations where a fixed number of people need to be seated in a row, the calculation is based on the number of people, such as 5! = 120 ways for 5 people to sit in a row.
• 😀 Permutation with partial elements applies when only a selected number of elements are used from a larger set. This is calculated as n! / (n-k)! where k is the number of selected elements.
• 😀 An example of permutation with partial elements is selecting 3 numbers from a set of 5 digits (1, 2, 3, 4, 5). The number of permutations is 5! / 2! = 60.
• 😀 Another example is choosing a president, treasurer, and secretary from a group of 6 people. This can be calculated as 6! / 3! = 120.
• 😀 Permutations emphasize the importance of order, unlike combinations where order does not matter.
• 😀 Permutation with identical elements requires dividing by the factorial of the repeated elements to avoid overcounting. For example, in the word 'Mathematics', the number of permutations is 10! / (2! * 2! * 2!) due to repeated letters.
• 😀 Circular permutations (permutations with cycles) involve arrangements where the order of elements in a circle matters. The formula is (n-1)! for n elements arranged in a circle.
• 😀 A practical example of circular permutation is determining how many ways 5 people can sit around a round table, calculated as (5-1)! = 24.

Q & A

What is permutation, and how is it calculated in the script?

Permutation refers to the arrangement of elements in a specific order. In the script, it's calculated using the factorial formula (n!) where n is the total number of elements, and the formula multiplies all integers from n down to 1.

What is the difference between permutation with all elements and permutation with some elements?

Permutation with all elements involves arranging every available element, while permutation with some elements involves selecting a subset of elements from a larger set and arranging them.

How is the permutation of all elements of the word 'Madiun' calculated?

Since 'Madiun' has 6 distinct letters, the permutation is calculated using 6 factorial (6!), which equals 720.

What is the formula for calculating permutation with a subset of elements?

The formula for permutation with a subset is n! / (n - k)!, where n is the total number of elements, and k is the number of elements to be selected and arranged.

How is the permutation of selecting 3 different numbers from 1 to 5 calculated?

The permutation of selecting 3 numbers from 1 to 5 is calculated as 5! / (5 - 3)! = 60.

How is permutation applied in the example of selecting a president, treasurer, and secretary from 6 people?

In this example, permutation is used because the roles (president, treasurer, and secretary) are specific and their order matters. The calculation is 6! / (6 - 3)! = 120.

What is the formula for calculating permutation when there are repeated elements?

The formula for permutation with repeated elements is n! / (p1! * p2! * ... * pk!), where n is the total number of elements, and p1, p2, ... pk represent the repeated elements.

How is the permutation of the word 'mathematics' calculated?

The word 'mathematics' has repeated letters. The permutation is calculated as 11! / (2! * 2! * 2! * 2!), where 11 is the total number of letters, and the factorials account for the repeated letters.

What is the formula for calculating circular permutation?

For circular permutation, the formula is (n - 1)!, where n is the number of elements to be arranged in a circle.

How is the permutation for 5 people sitting in a circle calculated?

The permutation for 5 people sitting in a circle is calculated as (5 - 1)! = 4!, which equals 24.