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 (nn) and the number of items to be arranged (rr), 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.