## Permutation

A permutation calculator is a tool designed to quickly and accurately compute the number of possible permutations of a set of items. It typically allows users to input the total number of items ($n$) and the number of items to be arranged ($r$), and then it uses the permutation formula to provide the result. Hereโs what a permutation calculator typically does:

#### Features of a Permutation Calculator

**1. Input Fields:**

โ Total Items (\( n \)): The number of items in the set.

โ Items to Arrange (\( r \)): The number of items to be arranged or selected from the set.

**2. Calculation:**

โ Uses the formula for permutations without repetition:

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

โ Computes the factorial of \( n \) and \( (n-r) \).

โ Divides \( n! \) by \( (n-r)! \) to get the number of permutations.

**3. Output:**

โ Displays the total number of permutations possible with the given \( n \) and \( r \).

#### Example Usage

Example 1: Calculate the number of permutations of 5 items taken 3 at a time.

**1. Input:**

โ Total Items (\( n \)) = 5

โ Items to Arrange (\( r \)) = 3

**2. Calculation:**

\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60 \]

**3. Output:**

โ The number of permutations is 60.

### Permutation: Detailed Tutorial

#### Introduction to Permutations

Definition: A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. In other words, permutations consider the arrangement and sequence of items.

#### Examples

1. For the set {A, B, C}, the permutations of 2 elements are:

โ AB

โ AC

โ BA

โ BC

โ CA

โ CB

2. For the set {1, 2, 3}, the permutations of all three elements are:

โ 123

โ 132

โ 213

โ 231

โ 312

โ 321

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#### Formula for Permutations

The formula to find the number of permutations of `n` items taken `r` at a time is given by:

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

Where:

โ \( n \) is the total number of items.

โ \( r \) is the number of items to be arranged.

โ \( n! \) (n factorial) is the product of all positive integers up to \( n \).

**Special Case**

โ When \( r = n \), the formula simplifies to:

\[ P(n, n) = n! \]

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#### Examples

1. Example 1: How many ways can you arrange 3 out of 5 books on a shelf?

โ Here, \( n = 5 \) and \( r = 3 \).

โ \( P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} \)

โ \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

โ \( 2! = 2 \times 1 = 2 \)

โ So, \( P(5, 3) = \frac{120}{2} = 60 \)

2. Example 2: How many different ways can you arrange all the letters of the word โCATโ?

โ Here, \( n = 3 \) and \( r = 3 \).

โ \( P(3, 3) = \frac{3!}{(3-3)!} = 3! = 6 \)

โ The permutations are: CAT, CTA, ACT, ATC, TCA, TAC

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#### Permutations with Repetition

When repetition is allowed, the number of permutations of `n` items taken `r` at a time is:

\[ P(n, r) = n^r \]

#### Example

โ If you have 3 types of ice cream (Vanilla, Chocolate, Strawberry) and you want to make a 2-scoop cone, where each scoop can be any of the 3 flavors, the number of possible permutations is:

\[ P(3, 2) = 3^2 = 9 \]

The possible permutations are: VV, VC, VS, CV, CC, CS, SV, SC, SS

#### Practice Problems

1. How many ways can you arrange 4 out of 7 different books on a shelf?

2. Find the number of permutations of the letters in the word โHOUSEโ.

3. How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if repetition is not allowed?

4. How many different 4-letter arrangements can be made from the word โBALLโ?

5. If repetition is allowed, how many 3-letter combinations can be made from the letters A, B, C?

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#### Answers to Practice Problems

1. \( P(7, 4) = \frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{5040}{6} = 840 \)

2. The word โHOUSEโ has 5 letters, all unique.

\[ P(5, 5) = 5! = 120 \]

3. The digits are 1, 2, 3, 4, and 5.

\[ P(5, 3) = \frac{5!}{2!} = \frac{120}{2} = 60 \]

4. The word โBALLโ has 4 letters, with โLโ repeating twice.

\[ P = \frac{4!}{2!} = \frac{24}{2} = 12 \]

5. With repetition allowed, 3-letter combinations from A, B, C:

\[ P(3, 3) = 3^3 = 27 \]

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#### Summary

โ Permutations consider the order of elements.

โ The formula for permutations without repetition is \( P(n, r) = \frac{n!}{(n-r)!} \).

โ For permutations with repetition, the formula is \( P(n, r) = n^r \).

โ Practice solving different types of permutation problems to become proficient in the concept.

Understanding permutations is fundamental in combinatorics and has numerous applications in mathematics, computer science, and everyday problem-solving.