Permutations

Given a set of n elements, the permutations are different groups of elements formed by the k elements of a subset such that:

The order of the elements does matter.

The elements are not repeated.

The permutations are denoted by P(n,k) or nPk.

Permutations

The permutations can also be calculated by factorials:

Permutations and Factorials

If k equals n:

Permutations

Permutations

Examples

1. Calculate P(6,3).

Permutations Exercise

Permutation Solution

2. Calculate P6.

P6 = 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720

3.How many different three-digit numbers can be formed with the digits: 1, 2, 3, 4, 5?

n = 5k = 3

The order of the elements does matter. The different numbers are 123, 231, 321...

The elements cannot be repeated as the problem states that the figures are different.

Permutation Example

4. How many different 5-digit numbers can be formed with the digits: 1, 2, 3, 4, 5?

n = 5     k = 5

The order of the elements does matter.

The elements cannot be repeated.

Permutations Exercise

5.How many different three-digit numbers can be formed with the digits: 0, 1, 2, 3, 4, 5?

n = 6k = 3

The numbers must be separated into two blocks:

bloques

The first set, of one number, can occupy only one of 5 digits because a number does not begin with zero (except for license plates and other special cases).

n = 5     k = 1

The second block, of two numbers, can occupy any digit less than the initial.

n = 5     k = 2

Permutation Solution

6. 10 candidates have been presented with awards for their novels in a literary contest. The honor roll (list of cadidates) is formed by the winner, finalist and runners-up. How many different honor rolls can be formed?

n = 10 k = 3

The order of the elements does matter as each honor roll cannot have the same winner and finalist.

Also, there are no repeated elements as it is assumed that each candidate has entered only one novel into the contest.

Permutations Solution

7. How many ways can eight people sit in a row of seats?

n = 8     k = 8

The order of the elements does matter.

There are no repeated elements as a person cannot be repeated.

Permutation Example