Permutations/Arrangements

Overview

A permutation is an arrangement of objects in a specific order. Permutations are used to count the number of possible arrangements when the order of selection matters.

Key Concepts

Factorial ($n!$):

The factorial of $n$ is the product of all positive integers up to $n$:

$$n! = n \times (n-1) \times (n-2) \times \ldots \times 1$$

Permutations of $n$ Distinct Objects:

The total number of arrangements of $n$ objects is given by:

$$n!$$

Permutations of $r$ Objects from $n$ Distinct Objects:

The number of ways to arrange $r$ objects from a set of $n$ distinct objects is:

$$P(n, r) = \frac{n!}{(n-r)!}$$

Permutations with Repetition:

If some objects are identical, the total permutations are reduced:

$$\text{Permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}$$

Where $p_1, p_2, \ldots, p_k$ are the counts of identical items.

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Worked Examples

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Summary

• Permutations calculate the number of arrangements when order matters.
• Formulas:
  • $n!$: Arrangements of n distinct objects.
  • $P(n, r) = \frac{n!}{(n-r)!}$: Arrangements of r objects from n distinct objects.
  • With repetition:

$$\frac{n!}{p_1! \times p_2! \times \ldots}$$

• Permutations are essential in counting problems and probability.