Permutations: Definition, Formulas and Example Problems
Permutations: Definition, Formulas and Example Problems – What does permutation mean and how do you calculate the math? On this occasion About the knowledge.co.id will discuss about Permutation and things about it. Let's look at the discussion together in the article below to better understand it.
Permutations: Definition, Formulas and Example Problems
An n factorial number notation is denoted by n! expresses the multiplication number n x (n-1) x (n-2) x (n-2) x … x 1, for example 7! = 7x6x5x4x3x2x1 = 5,040. This notation is used in calculations of permutations and combinations. Defined 0! = 1.
What is meant by permutation is the number of ways to make an arrangement with a certain number of members from the members of a set.
Permutation Formulas
The Ordinary Permutation Formula
For example, it is known that the set has n number of members, then the ordered arrangement consists of r the members are called permutations r of n, written as P(n, r) where r is less or equal with n. The permutation formula is as follows.
If r = n, then P(n, n) = n! (remember 0!=1)
An example for calculating the number of ways to arrange a sequence of two letters from the letters a, b, c is as follows:
The six ways are: ab, ac, ba, bc, ca, cb.
Equal Elements Permutation Formula
Suppose it is known that a set has n number of members, where there are n1 members of type 1 which the same, n2 members of the same type 2, etc., then the permutations of the members of the set are written as P(n; n1,n2,…,nk). The permutation formula if there are n1 members of the same type 1, n2 members of the same type 2, etc. is as follows.
An example for calculating the number of ways to arrange the letters of the word "KATAKKU" is as follows.
There are 3 letters K, so n1 = 3
There are 2 letters A, so n2 = 2
The letter T is 1, so n3 = 1
The letter U is 1, so n4 = 1
Cyclic Permutation Formulas
Cyclical permutations are permutations that are made by arranging elements in a circle according to a certain direction of rotation. very general. The problem is usually about the arrangement of people at the dinner table, conference table, etc.
The formula is simple: (n-1)!, where n is the number of objects/people present
Example: 5 directors sit at a circular table for a meeting. How many ways are there to arrange the chairs for the directors?
Answer: (5-1)! = 4! = 4 x 3 x 2 x 1 = 24
The Difference between Permutations and Combinations
The difference between permutations and combinations is that permutations pay attention to the order in which the members are arranged, while combinations do not pay attention to the order in which the members are arranged. This can be seen from the two examples above, namely the permutations and combinations of 2 members of the set consisting of the letters a, b, and c.
P(3,2) = 6 The six ways are: ab, ac, ba, bc, ca, cb.
C(3,2) = 3 The three ways are: ab, ac, bc.
Examples of Permutation Problems
Problem 1: 3 children will sit together on a long bench. In how many ways can they sit together on the bench?
Answer:
The three children will sit together, so the permutation formula P(3,3) is used
P(3,3) = 3! = 2x2x1 = 6
So the three children can sit together in 6 ways
Question 2: What is the permutation value of P(5,4) ?
a. 60
b. 80
c. 20
d. 22
Discussion: P(5,3)= 5!(5-3)!= 5.4.3.2!2! = 60
Answer: a
Question 3:The four officials who were invited came separately (not simultaneously). How many ways can the four officials come to =…?
a. 4
b. 8
c. 18
d. 12
Discussion:
It is known: n = 4, stating the number of officials invited r = 1,
states come independently P(4,1)= 4!(4-1)!= 4.3!3! = 4
Answer: a
Problem 4: A school will arrange a sports team consisting of 5 students who will be nominated to become players. But only 3 people can be the main player. Determine the number of ways that can be used to select the main players?
a. 60
b. 20
c. 90
d. 12
Discussion: It is known: n = 5, stating the number of students who will be nominated in the sports team r = 3, stating the number of students who may be the main players. P(5,3)= 5!(5-3)!= 5.4.3.2!2!2! = 60
Answer: a
Question 5:There are 5 chess players who will compete for first, second and third place in a chess tournament. How many first, second and third place orders can be formed from these five players?
From the problem above, we will make a sequence of 3 champions out of 5 chess players, so k = 3k=3 and n = 5n=5. When using the permutation formula, there are a number of winning arrangements that can be formed
Also Read: 1 Kg How Many Grams
Answer: nPk=5P3=(5−3)!5!=2!5!=3×4×5=60
Problem 6: How many ways are there to arrange the two letters of the word “LIFE”?
Answer:
How to arrange 2 letters out of 5 letters, then use the permutation P(5,2)
P(5,2) = (5!)/(5-2)! =(5x4x3!)/(3)! = 5×4 =20
So there are 20 ways to arrange the two letters of the word LIFE
Problem 7: How many ways are there to attach 5 banners along an alley consisting of 2 red banners and 3 yellow banners?
Answer:
The number of pennant pennant = 5 (2 red and 3 yellow), then the permutation using the same members P(5;2,3)
P(5;2,3) = (5!)/(2! 3!) = (5×4)/(2×1) = 10
So there are 10 ways to install banners
Problem 8: In a meeting attended by 12 participants. Each participant shakes hands. How many handshakes were there between them?
Answer:
Each handshake only involves 2 people, so the combination C(12,2) is used
C(12,2) = (12!)/(2!(12-2)!) = (12x11x10!)/(2x1x10!) = 66
Thus the review from About the knowledge.co.id about Permutation, hopefully can add to your insight and knowledge. Thank you for visiting and don't forget to read other articles.
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