Permutation is a mathematical concept that deals with the arrangement of objects in a specific order. For example, if you have three letters A, B, and C, you can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, and CBA. Each of these arrangements is called a permutation of the three letters.
Permutations are useful for counting the number of possible outcomes in various situations, such as ordering items, forming words, encrypting messages, and solving puzzles. Permutations are also related to the concept of combinations, which are selections of objects without regard to order.
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Permutation Formula without Repetition
To calculate the number of permutations of n objects taken r at a time without repetition, we can use the following formula:
where n! (read as n factorial) is the product of all positive integers from 1 to n and 0! is defined as 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
The formula can be understood as follows: To form a permutation of r objects from n objects, we have n choices for the first object, n-1 choices for the second object, n-2 choices for the third object, and so on, until we have n-r+1 choices for the r-th object. Multiplying these choices together gives us the numerator n! To avoid counting the same permutation more than once, we need to divide by the number of ways to arrange the r objects among themselves, which is r! However, since we do not have r! in the numerator, we use the equivalent expression (n-r)! instead.
For example, if we want to find the number of ways to arrange 4 letters out of the word MATH, we can use the formula:
Some special cases of the permutation formula without repetition are:
- When r = n, we have P(n,n) = n! which is the number of ways to arrange all n objects.
- When r = 1, we have P(n,1) = n, which is the number of ways to choose one object from n objects.
- When r = 0, we have P(n,0) = 1, which is the number of ways to choose no object from n objects.
Permutation Formula with Repetition
Sometimes, we may encounter situations where repetition of objects is allowed in permutations. For example, if we want to find the number of three-digit numbers that can be formed using the digits 0 to 9, we can use each digit more than once. In this case, we can use a different formula:
where n is the number of objects to choose from and r is the number of objects to choose. The formula can be explained as follows: To form a permutation of r objects from n objects with repetition, we have n choices for each object. Therefore, multiplying these choices together gives us nr.
For example, if we want to find the number of three-digit numbers that can be formed using the digits 0 to 9, we can use the formula:
Note that this formula does not account for any restrictions on the permutations. For instance, if we want to exclude numbers that start with zero or have repeated digits, we need to apply some additional logic.
Permutation is a way of arranging objects in a specific order. It can be calculated using different formulas depending on whether repetition is allowed or not. A permutation is an important concept in combinatorics and probability theory.
Permutation Examples and Applications
Permutations are used to model various scenarios where the order of objects matters. Here are some examples and applications of permutations:
Ordering items
If you have n items and you want to arrange them in a row, you can use permutations to find the number of possible arrangements. For example, if you have 5 books and you want to place them on a shelf, you can use the formula P(5,5) = 5! = 120 to find the number of ways to do so.
Forming words
If you have n letters and you want to form a word of length r, you can use permutations to find the number of possible words. For example, if you have the letters A, B, C, D, and E and you want to form a 3-letter word, you can use the formula P(5,3) = 60 to find the number of words. However, if some letters are repeated, you need to adjust the formula by dividing by the factorial of the number of repetitions. For example, if you have the letters A, A, B, C, and D and you want to form a 3-letter word, you can use the formula P(5,3) / 2! = 30 to find the number of words.
Encrypting messages
If you want to encrypt a message by replacing each letter with another letter according to a certain rule, you can use permutations to find the number of possible encryption schemes. For example, if you want to encrypt a message using a simple substitution cipher, where each letter is replaced by another letter in the alphabet, you can use the formula P(26,26) = 26! to find the number of possible ciphers.
Solving puzzles
If you want to solve a puzzle that involves arranging objects in a certain way, you can use permutations to find the number of possible solutions. For example, if you want to solve a Rubik’s cube, where each face has 9 stickers of one of six colors, you can use the formula P(54,54) / (8! x 3^8 x 12! x 2^12 x 6!) to find the number of possible configurations.
These are just some of the examples and applications of permutations. Permutations are also used in other fields such as computer science, biology, cryptography, music theory, and more.
Permutation Problems and Solutions
Permutation problems are problems that involve finding the number of ways to arrange a set of objects in a specific order. Here are some examples of permutation problems and their solutions:
How many ways can 6 people sit in a row of 6 chairs?
Solution: To arrange 6 people in a row of 6 chairs, we can use the permutation formula without repetition: P(6,6) = 6! = 720. Therefore, there are 720 ways to seat 6 people in a row of 6 chairs.
How many 4-letter words can be formed from the letters of the word BINGO?
Solution: To form a 4-letter word from the letters of BINGO, we can use the permutation formula without repetition: P(5,4) = 5 x 4 x 3 x 2 = 120. Therefore, there are 120 possible 4-letter words from BINGO.
How many different license plates can be made using three letters followed by three digits?
Solution: To make a license plate using three letters followed by three digits, we can use the permutation formula with repetition: P(26,3) x P(10,3) = 26^3 x 10^3 = 17576000. Therefore, there are 17576000 possible license plates with this format.
How many ways can a committee of 3 be chosen from a group of 10 people?
Solution: This is not a permutation problem, but a combination problem since the order of the committee members does not matter. To find the number of combinations of n objects taken r at a time, we can use the following formula:
Therefore, to choose a committee of 3 from a group of 10 people, we can use the formula: C(10,3) = (10 x 9 x 8) / (3 x 2 x 1) = 120. Therefore, there are 120 ways to form a committee of 3 from a group of 10 people.
Permutation vs Combination: What is the Difference?
Permutation and combination are two related concepts in mathematics that deal with the selection and arrangement of objects from a set. The main difference between them is that permutation considers the order of the objects, while combination does not.
For example, suppose you have a set of three letters: A, B, and C. You want to select two letters from this set and form a word. How many possible words can you make?
If you use permutation, you care about the order of the letters. For example, AB and BA are two different words. To find the number of permutations of n objects taken r at a time, you can use the formula:
where n! (read as n factorial) is the product of all positive integers from 1 to n, and 0! is defined as 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Using this formula, you can find the number of permutations of 3 letters taken 2 at a time:
Therefore, there are 6 possible words using permutation: AB, AC, BA, BC, CA, and CB.
If you use the combination, you do not care about the order of the letters. For example, AB and BA are the same words. To find the number of combinations of n objects taken r at a time, you can use the formula:
Using this formula, you can find the number of combinations of 3 letters taken 2 at a time:
Therefore, there are 3 possible words using combination: AB, AC, and BC.
In general, permutation is used when the order of the objects matters, while combination is used when the order does not matter. Permutation and combination are important concepts in combinatorics and probability theory.