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## Probability: Mastering Permutations and Combinations

Probability is the study of how likely something is to happen. It is a branch of mathematics that deals with uncertainty and randomness. Probability can help us make predictions, decisions, and inferences based on data and experiments.

One of the fundamental concepts in probability is the idea of counting. Counting is the process of finding out how many possible outcomes there are for a given situation. For example, if we toss a coin, how many ways can it land? If we roll a die, how many numbers can we get? If we draw a card from a deck, how many suits and ranks are there?

Counting can be simple or complex, depending on the scenario. Sometimes, we need to consider the order of the outcomes, such as when we arrange objects in a line or select items from a list. Other times, we only care about the number of outcomes, regardless of their order, such as when we form groups or subsets from a collection.

Permutations and combinations are two techniques that help us count the number of outcomes in different situations. They are based on the use of factorials, which are products of consecutive positive integers. For example, 5! (read as "five factorial") is equal to 5 x 4 x 3 x 2 x 1 = 120.

## Permutations

A permutation is an ordered arrangement of objects. For example, if we have three letters A, B, and C, we can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, CBA. Each arrangement is a permutation of the three letters.

To find the number of permutations of n objects, we can use the formula n!. For example, to find the number of permutations of 5 objects, we can use 5! = 120.

However, sometimes we only want to arrange some of the objects, not all of them. For example, if we have 10 books and we want to arrange 3 of them on a shelf, how many ways can we do that? In this case, we need to use another formula: nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects we want to arrange. For example, to find the number of ways to arrange 3 books out of 10, we can use 10P3 = 10! / (10 - 3)! = 720.

## Combinations

A combination is an unordered selection of objects. For example, if we have three letters A, B, and C, we can select two of them in three different ways: AB, AC, BC. Each selection is a combination of the three letters.

To find the number of combinations of n objects taken r at a time, we can use the formula nCr = n! / (r! x (n - r)!), where n is the total number of objects and r is the number of objects we want to select. For example, to find the number of ways to select 2 letters out of 3, we can use 3C2 = 3! / (2! x (3 - 2)!) = 3.

The formula for combinations can also be written as nCr = nPr / r!, where nPr is the number of permutations of n objects taken r at a time. This shows that combinations are related to permutations by dividing out the order. For example, to find the number of ways to select 2 letters out of 3, we can also use 3C2 = 3P2 / 2! = 6 / 2 = 3.

## Applications

Permutations and combinations are useful tools for solving various problems in probability and statistics. For example:

• If we want to find the probability of getting a certain hand in poker, we need to count how many possible hands there are and how many hands match our criteria.

• If we want to find the probability of winning a lottery, we need to count how many possible tickets there are and how many tickets match the winning numbers.

• If we want to find the probability of forming a committee from a group of people, we need to count how many ways there are to choose the members and how many ways meet our requirements.

## Resources

If you want to learn more about permutations and combinations and how to apply them in probability problems, you can check out these resources:

• [18.600 F2019 Lecture 1: Permutations and combinations]: This is a lecture note from MIT OpenCourseWare that covers the basics of counting, factorials, permutations, and combinations.

• [FACTORIALS, PERMUTATIONS AND COMBINATIONS]: This is a PDF document from Florida State University that explains the concepts and formulas of factorials, permutations, and combinations with examples and exercises.

• [MATH 106 Lecture 2 Permutations & Combinations]: This is a PDF document from Michigan State University that illustrates the use of permutations and combinations in probability problems with examples and solutions.

## If you want to download these resources as PDF files, you can use the links provided above or use a web browser extension that allows you to save web pages as PDF files. Practice

Now that you have learned the basics of permutations and combinations, you can practice your skills by solving some problems. Here are some examples of questions that you can try to answer using the formulas and techniques that you have learned:

• How many different ways can you arrange the letters in the word "BING"?

• How many different four-digit codes can you make using the digits 0 to 9, if each digit can only be used once?

• How many different ways can you choose 3 toppings for your pizza from a list of 10 available toppings?

• How many different ways can you form a committee of 5 people from a group of 12 people?

• How many different ways can you distribute 10 identical balls into 4 distinct boxes?

• There are 4! = 24 ways to arrange the letters in the word "BING".

• There are 10P4 = 10! / (10 - 4)! = 5040 ways to make a four-digit code using the digits 0 to 9, if each digit can only be used once.

• There are 10C3 = 10! / (3! x (10 - 3)!) = 120 ways to choose 3 toppings for your pizza from a list of 10 available toppings.

• There are 12C5 = 12! / (5! x (12 - 5)!) = 792 ways to form a committee of 5 people from a group of 12 people.

• There are (10 + 4 - 1)C(4 - 1) = 13C3 = 13! / (3! x (13 - 3)!) = 286 ways to distribute 10 identical balls into 4 distinct boxes.

If you want more practice problems, you can use these resources:

• [Permutations and Combinations Practice Problems]: This is a web page from Math Is Fun that provides interactive practice problems on permutations and combinations with hints and solutions.

• [Permutations and Combinations Worksheet]: This is a PDF document from Kuta Software that contains printable worksheets on permutations and combinations with answers.

• [Permutations and Combinations Quiz]: This is a web page from Softschools.com that allows you to take a quiz on permutations and combinations with instant feedback.

## Conclusion

Permutations and combinations are powerful tools that can help us count the number of possible outcomes in various situations involving probability and statistics. They can also help us understand the concepts of sampling, distribution, and hypothesis testing. By mastering these techniques, we can improve our analytical and critical thinking skills and solve complex problems more efficiently.

If you want to download this article as a PDF file, you can use the link below or use a web browser extension that allows you to save web pages as PDF files.