Review on Permutations
The number of different ways in which the elements of a set can be arranged is called a permutation. In a permutation the order of the groups matters.
If the elements in the set are not repeated, the total number of ways in which m elements can be placed in groups of size n is given by the formula:
If in a given case m=n, to calculate the total number of permutations the following formula is used:
If the elements in the set are repeated, that is, the first element is repeated 
times, the second 
times and so on, being 
, the number of permutations taking into account the repeated elements is obtained with the formula:
Permutation Exercises
How many 5-digit numbers with different digits can be formed with the digits: 1, 2, 3, 4, 5?
1 We establish the conditions of the exercise:
All elements are included.
Order matters.
Elements are not repeated. The problem asks us for the digits to be different
2 It is a Permutation so we use the formula:
3 We substitute and solve:
In how many different ways can eight people sit in a row of seats?
1 We establish the conditions of the exercise:
All elements are included. 8 people must sit
Order matters
Elements are not repeated. A person cannot be repeated
2 It is a Permutation so we use the formula:
3 We substitute and solve:
In how many different ways can eight people sit around a round table?
1 We establish the conditions of the exercise:
All elements are included. 8 people must sit
Being a circular arrangement we must eliminate circular repetitions
Elements are not repeated. A person cannot be repeated
2 It is a Combination so we use the formula:
3 We substitute and solve:
With the digits 2, 2, 2, 3, 3, 3, 3, 4, 4; how many nine-digit numbers can be formed?
1 We have 3 elements a, b, c, that are repeated:
2 It is a permutation with several elements that are repeated so we use the formula:
3 We substitute into the formula and solve:
With the letters of the word "piano". How many different arrangements can be made that start with a vowel?
1 The word starts with i, a, or o followed by the 4 remaining letters taken 4 at a time.
2 We establish the conditions:
All elements are included
Order matters
Elements are not repeated
3 Having 3 vowels with which it can start we can calculate the result with:
How many five-digit numbers with different digits can be formed with odd digits? How many of them are greater than 70,000?
1 We establish the conditions:
All elements are included
Order matters
Elements are not repeated
2 To find the five-digit numbers using odd digits (1,3,5,7,9) we use the following formula:
We substitute and solve:
To find the numbers that are greater than 70,000 we consider those that start with 7 or 9
On a ship's signal mast, three red flags, two blue flags and four green flags can be hoisted. How many different signals can be indicated with the placement of the nine flags?
1 We have 3 elements a, b, c, that are repeated:
2 It is a permutation with several elements that are repeated so we use the formula:
3 We substitute into the formula and solve:
In how many ways can the 11 players of a soccer team be positioned taking into account that the goalkeeper cannot occupy any position other than the goal?
1 We establish the conditions of the exercise:
We have 10 players who can occupy 10 different positions.
All elements are included
Order matters
Elements are not repeated
2 It is a Permutation so we use the formula:
3 We substitute and solve:
A head table is made up of eight people. In how many different ways can they sit, if the president and the secretary always go together?
1 We consider the two people who must go together as one person which can be achieved in 2! ways. Now there are seven people to seat around the table and it is true that:
All elements are included
Order matters
Elements are not repeated
2 We can solve the exercise with:
Four different mathematics books, six different physics books and two different chemistry books are placed on a shelf. In how many different ways can they be arranged if:
1 The books of each subject must all be together.
2 Only the mathematics books must be together.
1 The books of each subject must all be together.
2 Only the mathematics books must be together.
Five red balls, 2 white balls and 3 blue balls are arranged in a row. If balls of the same color are indistinguishable from each other, in how many possible ways can they be arranged?
1 We have 3 elements a, b, c, that are repeated:
2 It is a permutation with several elements that are repeated so we use the formula:
3 We substitute into the formula and solve:
Solve the equations
1 
2 
3 
1
We discard the negative solution so 
2
We discard the negative solution so 
3
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