PERMUTASI dari Unsur Berbeda, Unsur Sama dan Siklis

Summary

This video tutorial delves into the concept of permutations in mathematics, covering three main types: permutations of different elements, permutations of identical elements, and cyclic permutations. The presenter explains key formulas, such as how to calculate the number of possible arrangements for distinct items and items that repeat. Additionally, the video includes problem-solving examples to illustrate the application of these concepts, like determining the number of ways to arrange individuals in specific roles and calculating arrangements of letters from a word with repeating characters. The final segment touches on cyclic permutations, especially in scenarios like seating arrangements in circular setups.

Takeaways

• 😀 Permutation is about arranging distinct elements in a specific order, considering the sequence.
• 😀 Permutation formulas are based on the number of elements (n) and the number of selected elements (k).
• 😀 In permutations of different elements, the formula is P(n, k) = n! / (n-k)!. Example: P(12, 3) = 1320.
• 😀 Permutations are also used in word arrangement problems, where repeated elements are accounted for in the formula.
• 😀 In permutations of identical elements, the formula adjusts for identical items by dividing by the factorial of repeated elements.
• 😀 Example of permutations with identical elements: arranging the word 'BERJEJER' results in 1680 distinct words.
• 😀 Permutations can also apply to seating arrangements, where the order matters but rotation does not (cyclic permutations).
• 😀 In cyclic permutations, the formula is P_c(n) = (n-1)!.
• 😀 Example of cyclic permutation: 7 participants in a round-table discussion have 720 possible seating arrangements.
• 😀 Understanding combinations is important to distinguish between problems requiring permutations (order matters) and combinations (order doesn't matter).
• 😀 The importance of the order in permutations is emphasized in problems where the roles or positions (e.g., president, vice president) are specified.

Q & A

What is the main focus of the video script?

The video script focuses on explaining the concept of permutations, covering three types: permutations of distinct elements, permutations of identical elements, and cyclic permutations, with examples and formulas.

What is a permutation?

A permutation is the number of ways to arrange a set of objects in a specific order, without repetition. The order matters in a permutation.

What distinguishes permutations from combinations?

In permutations, the order of elements matters, while in combinations, the order does not matter.

What is the formula for calculating permutations of distinct elements?

The formula for permutations of distinct elements is P(n, k) = n! / (n - k)!, where 'n' is the total number of objects and 'k' is the number of objects to be arranged.

Can you explain the example where 3 objects are selected from 12?

In this example, we calculate P(12, 3) using the formula P(12, 3) = 12! / (12 - 3)! = 12 × 11 × 10 = 1320.

What is the formula for permutations with identical elements?

For permutations with identical elements, the formula is P(n, k1, k2, ..., kn) = n! / (k1! × k2! × ... × kn!), where k1, k2, ..., kn are the counts of identical elements.

How do you calculate the number of permutations of the word 'berjejer'?

The word 'berjejer' has the following counts of identical letters: 'e' appears 3 times, 'r' appears 2 times, and 'j' appears 2 times. The number of permutations is P(8, 3, 2, 2) = 8! / (3! × 2! × 2!) = 1680.

What is a cyclic permutation?

A cyclic permutation refers to the number of ways to arrange objects in a circular arrangement, where rotating the objects does not result in a new arrangement.

How do you calculate the number of cyclic permutations for 7 participants around a round table?

For 7 participants, the number of cyclic permutations is calculated using the formula P_cyclic(n) = (n - 1)!, so P_cyclic(7) = 6! = 720.

What key concept is emphasized when calculating permutations of identical elements?

The key concept in calculating permutations of identical elements is recognizing that identical elements do not create new distinct arrangements and must be accounted for by dividing by the factorial of their counts.