# 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 _{n}**P _{k}**.

The permutations can also be calculated by factorials:

If **k** equals **n**:

#### Examples

1. Calculate P(6,3).

2. Calculate P_{6}.

P_{6} = 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.

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.

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:

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

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.

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.