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Master Permutation and Combination with this Free PDF Guide



Aptitude Questions on Permutation and Combination PDF Free




Are you preparing for an aptitude test that involves permutation and combination questions? If so, you might be wondering how to master this topic and ace the test. In this article, we will provide you with everything you need to know about permutation and combination, including definitions, formulas, examples, tips, tricks, common mistakes, and a free PDF file with practice questions. By the end of this article, you will be able to solve any permutation and combination problem with confidence and ease.




aptitude questions on permutation and combination pdf free



What are permutation and combination?




Permutation and combination are two ways of counting the number of different arrangements or selections that can be made from a given set of objects. They are often used in probability, statistics, combinatorics, cryptography, and other fields that deal with patterns and arrangements.


A permutation is an ordered arrangement of r objects from a set of n objects. For example, if you have three letters A, B, and C, you can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, CBA. Each of these arrangements is a permutation of three letters from a set of three letters.


A combination is an unordered selection of r objects from a set of n objects. For example, if you have three letters A, B, and C, you can select two of them in three different ways: AB, AC, BC. Each of these selections is a combination of two letters from a set of three letters.


Why are they important for aptitude tests?




Permutation and combination questions are common in aptitude tests because they test your logical thinking, numerical reasoning, and problem-solving skills. They also require you to apply basic mathematical concepts such as factorial, multiplication, addition, division, subtraction, etc. Permutation and combination questions can appear in various forms, such as finding the number of possible arrangements or selections, finding the probability of a certain event or outcome, finding the number of ways to perform a task or achieve a goal, etc.


Permutation and combination questions can be challenging if you are not familiar with the concepts and formulas involved. However, if you understand the logic behind them and practice enough problems, you will find them easy and fun to solve.


How to Solve Permutation and Combination Problems




The first step to solve any permutation or combination problem is to identify whether it is a permutation or a combination problem. A simple way to do this is to ask yourself whether the order of arrangement or selection matters or not. If it does matter, then it is a permutation problem. If it does not matter, then it is a combination problem.


The next step is to apply the appropriate formula or rule to find the answer. Here are some basic formulas and rules that you should know:


Basic formulas and rules




  • The factorial of a positive integer n, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.



  • The number of permutations of n objects taken r at a time, denoted by nPr, is given by nPr = n! / (n - r)!. For example, the number of permutations of 5 objects taken 3 at a time is 5P3 = 5! / (5 - 3)! = 120 / 2! = 60.



  • The number of combinations of n objects taken r at a time, denoted by nCr, is given by nCr = n! / (r! x (n - r)!). For example, the number of combinations of 5 objects taken 3 at a time is 5C3 = 5! / (3! x (5 - 3)!) = 120 / (6 x 2) = 10.



  • If there are n objects of which p are alike and q are alike and the rest are distinct, then the number of permutations of these objects is given by n! / (p! x q!). For example, if there are 6 letters of which 2 are A and 2 are B and the rest are distinct, then the number of permutations of these letters is 6! / (2! x 2!) = 360 / (2 x 2) = 90.



  • If there are n objects of which p are alike and the rest are distinct, then the number of combinations of these objects taken r at a time is given by (n - p + 1)Cr. For example, if there are 6 letters of which 2 are A and the rest are distinct, then the number of combinations of these letters taken 3 at a time is (6 - 2 + 1)C3 = 5C3 = 10.



Examples and solutions




Here are some examples of permutation and combination problems and their solutions:



Problem


Solution


How many different four-digit numbers can be formed using the digits 1, 2, 3, and 4 without repetition?


This is a permutation problem because the order of the digits matters. We have to arrange four digits from a set of four digits. The number of permutations is given by 4P4 = 4! / (4 - 4)! = 24 / 1 = 24.


How many ways can a committee of three be chosen from a group of nine people?


This is a combination problem because the order of the people does not matter. We have to select three people from a set of nine people. The number of combinations is given by 9C3 = 9! / (3! x (9 - 3)!) = 504 / (6 x 6) = 28.


How many different words can be formed using the letters of the word "APPLE"?


This is a permutation problem because the order of the letters matters. However, we have to consider that there are two A's and two P's in the word. The number of permutations is given by 5! / (2! x 2!) = 30.


How many ways can five books be arranged on a shelf if two of them are identical?


This is a permutation problem because the order of the books matters. However, we have to consider that two books are identical. The number of permutations is given by 5! / (2!) = 60.


How many ways can four red balls and two blue balls be arranged in a row?


This is a combination problem because the order of the balls does not matter. However, we have to consider that there are four red balls and two blue balls that are identical. The number of combinations is given by (6 - 4 + 1)C2 = (6 - 2 + 1)C4 = 15.


Tips and Tricks for Permutation and Combination Questions




Tips and Tricks for Permutation and Combination Questions




Besides knowing the basic formulas and rules, here are some tips and tricks that can help you solve permutation and combination questions faster and easier:


Use factorial notation




Factorial notation is a convenient way to write large numbers that are products of consecutive integers. For example, instead of writing 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, you can write 8!. This can save you time and space when calculating permutations and combinations.


However, you should also be familiar with some common factorial values, such as 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, etc. This can help you simplify some expressions and avoid using a calculator.


Use the nCr and nPr functions on your calculator




Most scientific calculators have built-in functions for calculating permutations and combinations. These are usually denoted by nCr and nPr, where n is the number of objects and r is the number of objects taken at a time. For example, to calculate 10C4, you can press 10 nCr 4 on your calculator and get the answer as 210.


Using these functions can save you time and effort when dealing with large numbers or complex expressions. However, you should also be careful not to make mistakes when entering the values or choosing the correct function.


Apply the multiplication and addition principles




The multiplication principle states that if there are m ways to do one thing and n ways to do another thing, then there are m x n ways to do both things. For example, if there are 5 shirts and 4 pants to choose from, then there are 5 x 4 = 20 ways to choose a shirt and a pant.


The addition principle states that if there are m ways to do one thing and n ways to do another thing that are mutually exclusive (i.e., they cannot happen at the same time), then there are m + n ways to do either thing. For example, if there are 5 shirts and 4 pants to choose from, then there are 5 + 4 = 9 ways to choose either a shirt or a pant.


These principles can help you break down complex problems into simpler ones and find the number of ways to do multiple tasks or achieve multiple outcomes.


Use Venn diagrams and set theory




Venn diagrams are graphical representations of sets and their relationships. They can help you visualize the number of elements in different sets and their intersections or unions. For example, if there are 20 students in a class, of which 12 like math and 8 like science, then you can draw a Venn diagram like this:



------------------ -------- M --------------- S ------------------


In this diagram, M represents the set of students who like math, S represents the set of students who like science, and the overlapping region represents the set of students who like both math and science. The number of elements in each region can be found by using the formula:


n(M S) = n(M) + n(S) - n(M S)


where n(M S) is the number of elements in the union of M and S (i.e., the students who like math or science or both), n(M) is the number of elements in M (i.e., the students who like math), n(S) is the number of elements in S (i.e., the students who like science), and n(M S) is the number of elements in the intersection of M and S (i.e., the students who like both math and science).


In this example, we have:


n(M S) = 20


n(M) = 12


n(S) = 8


n(M S) = ?


Plugging these values into the formula, we get:


20 = 12 + 8 - n(M S)


n(M S) = 0


This means that there are no students who like both math and science in this class.


Venn diagrams and set theory can help you solve problems that involve overlapping or disjoint sets, such as finding the number of ways to choose from different categories or groups.


Common Mistakes to Avoid in Permutation and Combination Questions




Here are some common mistakes that you should avoid when solving permutation and combination questions:


Confusing permutation and combination




As mentioned earlier, permutation and combination are different ways of counting the number of arrangements or selections. Permutation considers the order of arrangement or selection, while combination does not. For example, if you have three letters A, B, and C, then AB and BA are two different permutations but one combination of two letters.


To avoid confusing permutation and combination, you should always ask yourself whether the order matters or not. If it does matter, then use permutation. If it does not matter, then use combination.


Forgetting to consider the order of arrangement or selection




Sometimes, the order of arrangement or selection matters only in some cases but not in others. For example, if you have to arrange four books on a shelf, then the order matters. However, if you have to choose two books from the shelf, then the order does not matter.


To avoid forgetting to consider the order of arrangement or selection, you should always read the question carefully and identify what exactly you have to arrange or select. If you have to arrange something, then use permutation. If you have to select something, then use combination.


Overcounting or undercounting the possibilities




Sometimes, you might count some possibilities more than once or miss some possibilities altogether. For example, if you have to choose two letters from A, B, C, and D, then you might count AB twice (once as AB and once as BA) or miss CD altogether.


To avoid overcounting or undercounting the possibilities, you should always check your answer for consistency and completeness. You can also use different methods to verify your answer, such as listing all the possibilities or using complementary counting.


Practice Permutation and Combination Questions with PDF Free Download




Now that you have learned the concepts, formulas, tips, tricks, and common mistakes of permutation and combination questions, you are ready to practice them with a free PDF file that contains 50 questions with answers and explanations. This PDF file will help you test your knowledge and skills on this topic and prepare for any aptitude test that involves permutation and combination questions.


How to download the PDF file




To download the PDF file for free, simply click on this link: https://www.example.com/aptitude-questions-on-permutation-and-combination-pdf-free. You will be redirected to a page where you can enter your name and email address. After entering your details, you will receive an email with a link to download the PDF file. You can also access the PDF file online without downloading it.


How to use the PDF file for practice




To use the PDF file for practice, you can either print it out or view it on your computer or mobile device. The PDF file contains 50 questions with four options each. You should try to solve each question within a given time limit (usually 1 minute) without using a calculator. After solving each question, you can check the answer and explanation at the end of the PDF file. You can also keep track of your score and progress by marking the questions that you got right or wrong.


Benefits of practicing with the PDF file




Practicing with the PDF file will provide you with several benefits, such as:


  • Improving your speed and accuracy in solving permutation and combination questions.



  • Enhancing your logical thinking, numerical reasoning, and problem-solving skills.



  • Familiarizing yourself with different types and formats of permutation and combination questions.



  • Boosting your confidence and performance in any aptitude test that involves permutation and combination questions.



Conclusion




Conclusion




In this article, we have covered everything you need to know about aptitude questions on permutation and combination pdf free. We have explained what permutation and combination are, why they are important for aptitude tests, how to solve permutation and combination problems, tips and tricks for permutation and combination questions, common mistakes to avoid in permutation and combination questions, and how to practice permutation and combination questions with a free PDF file. We hope that this article has helped you understand and master this topic and prepare for any aptitude test that involves permutation and combination questions.


If you have any questions or feedback, please feel free to leave a comment below. We would love to hear from you and help you out. Also, if you found this article useful, please share it with your friends and colleagues who might benefit from it. Thank you for reading and happy practicing!


FAQs




Here are some frequently asked questions and answers about aptitude questions on permutation and combination pdf free:



  • What is the difference between permutation and combination?



A permutation is an ordered arrangement of r objects from a set of n objects. A combination is an unordered selection of r objects from a set of n objects.


  • How do I know whether to use permutation or combination?



You can use permutation if the order of arrangement or selection matters. You can use combination if the order of arrangement or selection does not matter.


  • What are some common formulas for permutation and combination?



Some common formulas are:


  • nPr = n! / (n - r)!



  • nCr = n! / (r! x (n - r)!)



  • n! / (p! x q!) if there are n objects of which p are alike and q are alike



  • (n - p + 1)Cr if there are n objects of which p are alike



  • What are some tips and tricks for permutation and combination questions?



Some tips and tricks are:


  • Use factorial notation



  • Use the nCr and nPr functions on your calculator



  • Apply the multiplication and addition principles



  • Use Venn diagrams and set theory



  • What are some common mistakes to avoid in permutation and combination questions?



Some common mistakes are:


  • Confusing permutation and combination



  • Forgetting to consider the order of arrangement or selection



  • Overcounting or undercounting the possibilities



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