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7.4: General Combinatorics Problems

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• Richard W. Beveridge
• Clatsop Community College

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As is often the case, practicing one type of problem at a time is helpful to master the techniques necessary to solve that type of problem. More difficult is deciding which type of problem you are working on and choosing which techniques to use to solve the problem. In this section, the problem types from the previous three sections are combined. EXERCISES 4.4 SET I 1) How many strings of six lower case letters from the English alphabet contain \(\quad\) a) the letter \(a ?\) \(\quad\) b) the letters \(a\) and \(b\) in consecutive positions with \(a\) preceding \(b\), with all the letters distinct? \(\quad\) c) the letters \(a\) and \(b\), where \(a\) is somewhere to the left of \(b\) in the string, with all the letters distinct? 2) Seven women and nine men are on the faculty in the mathematics department at a school. \(\quad\) a) How many ways are there to select a committee of five members of the department if at least one woman must be on the committee? \(\quad\) b) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee? 3) Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have the same number of men and women? 4) The English alphabet contains 21 consonants and 5 vowels. How many strings of 6 lower case letters of the English alphabet contain: \(\quad\) a) exactly one vowel? \(\quad\) b) exactly 2 vowels? \(\quad\) c) at least 1 vowel? \(\quad\) d) at least 2 vowels? 5) How many ways are there to select 12 countries in the United Nations to serve on a council if 3 are selected from a block of 45,4 are selected from a block of \(57,\) and the others are selected from the remaining 69 countries? 6) Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have more women than men? 7) How many license plates consisting of three letters followed by three digits contain no letter or digit twice? 8) In the women's tennis tournament at Wimbledon, two finalists, A and B, are competing for the title, which will be awarded to the first player to win two sets. In how many different ways can the match be completed? 9) In the men's tennis tournament at Wimbledon, two finalists, \(A\) and \(B\), are competing for the title, which will be awarded to the first player to win three sets. In how many different ways can the match be completed? 10) In how many different ways can a panel of 12 jurors and 2 alternate jurors be chosen from a group of 30 prospective jurors?

SET II 11) A class has 20 students, of which 12 are female and 8 are male. In how many ways can a committee of five students be picked from the class if: \(\quad\) a) No restrictions are placed on the choice of students \(\quad\) b) No males are included on the committee \(\quad\) c) The committee must have three female members and 2 male members 12) A school dance committee is to be chosen from a group of students consisting of six freshman, eight sophomores, twelve juniors and ten seniors. If the committee should consist of two freshman, three sophomores, four juniors and five seniors, how many ways can this be done? 13) A theater company consists of 22 actors - 10 men and 12 women. In the next play, the director needs to cast a leading man, leading lady, supporting male role, supporting female role and eight extras (three men and five women). In how many ways can this play be cast? 14) A hockey coach has 20 players of which twelve play forward, six play defense and two are goalies. In how many ways can the coach pick a lineup consisting of three forwards, two defense players and one goalie? 15) In how many ways can ten students be arranged in a row for a class picture if John and Jane want to stand next to each other and Ed and Sally also want to stand next to each other? 16) In how many ways can the students in the previous problem be arranged if Ed and Sally want to stand next to each other, but John and Jane refuse to stand next to each other? 17) In how many ways can four men and four women be seated in a row of eight seats if: \(\quad\) a) The first seat is occupied by a man \(\quad\) b) The first and last seats are occupied by women

SET III 18) The social security number of a person is a sequence of nine digits that are not necessarily distinct. How many social security numbers are possible? 19) A variable name in the BASIC programming language is either a letter of the alphabet or a letter followed by a digit. How many distinct variable names are there in the BASIC language? 20) a) How many even numbers are there between 0 and 100? b) How many even numbers with distinct digits are there between 0 and \(100 ?\) 21) There are six characters - three letters of the English alphabet followed by three digits - which appear on the back panel of a particular brand of printer as an identification number. Find the number of possible identification numbers if \(\quad\) a) characters can be repeated \(\quad\) b) digits cannot repeat \(\quad\) c) letters cannot repeat \(\quad\) d) characters cannot repeat 22) A sequence of characters is called a palindrome if it reads the same forwards and backwards. For example \(\mathrm{K} 98 \mathrm{EE} 89 \mathrm{K}\) is an eight character palindrome and \(\mathrm{K} 98 \mathrm{E} 89 \mathrm{K}\) is a seven character palindrome. A MAN A PLAN A CANAL PANAMA is also a palindrome as are WAS IT A RAT I SAW, TACO CAT, TANGY GNAT, and NEVER ODD OR EVEN. Find the number of nine character palindromes that can be formed using the letters of the alphabet such that no letter appears more than twice in each one. 23) Find the number of ways to form a four-letter sequence using the letters \(A\), B, \(C, D\) and \(E\) if: \(\quad\) a) repetitions are permitted \(\quad\) b) repetitions are not permitted \(\quad\) c) the sequence contains the letter \(A\) but repetitions are not permitted \(\quad\) d) the sequence contains the letter \(A\) and repetitions are permitted 24) There are 10 members A, B, C, D, E, F, G, H, I and J in a fund raising committee. The first task of the committee is to choose a chairperson, a secretary and a treasurer from this group. No individual can hold more than one office. How many ways can these three positions be filled if: \(\quad\) a) no one has any objection for holding any of these offices \(\quad\) b) Cneeds to be the chairperson \(\quad\) c) \(\quad\) B does not want to be the chairperson \(\quad\) d) \(\quad\) A is not willing to serve as chairperson or secretary \(\quad\) e) either I or J must be the treasurer \(\quad\) f) \(\quad\) E or \(F\) or \(G\) must hold one of the three offices 25) Find the number of ways of picking each of the following from a standard deck of cards. \(\quad\) a) a king and a queen \(\quad\) b) a king or a queen \(\quad\) c) a king and a red card \(\quad\) d) a king or a red card 26) There are three bridges connecting two towns A and B. Between towns B and C there are four bridges. A salesperson needs to travel from A to C via B. Find: \(\quad\) a) the number of possible choices of bridges from A to \(C\) \(\quad\) b) the number of choices for a round trip from A to C and back to A \(\quad\) c) the number of choices for a round trip if no bridge may be crossed twice 27) A sequence of digits in which each digit is 0 or 1 is called binary number. Eight digit binary numbers are often referred to as "bytes." \(\quad\) a) How many bytes are possible? \(\quad\) b) How many bytes begin with 10 and end with \(01 ?\) \(\quad\) c) How many bytes begin with 10 but do not end with \(01 ?\) \(\quad\) d) How many bytes begin with 10 or end with \(01 ?\) 28) A group of 12 is to be seated in a row of chairs. How many ways can this be done if: \(\quad\) a) two people, \(A\) and \(B\) must be seated next to each other? \(\quad\) b) two people \(A\) and \(B\) must not be seated next to each other? 29) A variable in the FORTRAN language is a sequence that has at most six characters with the first character being a letter of the alphabet and the remaining characters (if any) being either letters or digits. How many distinct variable names are possible? 30) Four station wagons, five sedans and six vans are to be parked in a row of 15 spaces. Find the nubmer of ways to park the vehicles if: \(\quad\) a) the station wagons are parked at the beginning, then the sedans, then the vans \(\quad\) b) vehicles of the same type must be parked together

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

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• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Problem-Solving Strategies for solving Combinatorics Assignments

Embarking on the journey of conquering combinatorics assignments requires more than just mathematical prowess. Combinatorics, a captivating branch of mathematics dealing with the art of counting and arrangement, demands a strategic approach and a solid foundation in various concepts. As you embark on your journey to tackle assignments in combinatorics, it's crucial to have a solid understanding of fundamental topics and effective problem-solving strategies. In this blog, we will delve into the critical topics that lay the groundwork for successful combinatorics assignments and explore effective strategies to help with your Combinatorics assignment in this intricate field.

Understanding the Basics of Combinatorics

Before delving into more complex concepts, it's essential to grasp the foundational ideas in combinatorics. These concepts serve as building blocks for solving more intricate problems later on. Here are a few key topics you should familiarize yourself with:

Permutations and Combinations

Permutations involve arranging objects in a specific order, while combinations focus on selecting objects without considering the order. Understanding the formulas for permutations and combinations is crucial.

The Multiplication Principle

The multiplication principle, also known as the counting principle, helps you count the total number of outcomes for multiple tasks performed sequentially. It's a fundamental concept that underlies many combinatorial problems.

The Addition Principle

The addition principle allows you to count the total number of outcomes for tasks that can be performed in different ways or cases. It's particularly useful for solving problems involving choices or alternatives.

Advanced Topics for Combinatorics Assignments

Once you have a solid grasp of the basics, you can move on to more advanced topics that will enhance your ability to tackle complex combinatorics assignments.

Binomial Theorem and Pascal's Triangle

Understanding the binomial theorem and Pascal's Triangle can be immensely beneficial. These concepts are often used to expand binomial expressions and calculate combinations efficiently.

Inclusion-Exclusion Principle

The inclusion-exclusion principle helps solve problems involving multiple sets or conditions. It's a powerful tool for counting elements that satisfy certain criteria while considering overlaps.

Generating Functions

Generating functions provide a formal way to represent sequences of numbers, making it easier to solve problems involving recurrence relations and counting.

Pigeonhole Principle

The pigeonhole principle might seem intuitive, but it has deep implications in combinatorics. It helps you recognize when there must be repetitions or patterns in a given set of objects.

Strategies for Solving Combinatorics Assignments

Equipped with a solid understanding of the essential and advanced topics in combinatorics, let's dive into strategies for approaching and solving assignments effectively.

Understand the Problem Statement

Understanding the problem statement is the crucial first step in navigating the world of combinatorics assignments. It involves dissecting the given scenario, deciphering the information provided, and grasping the specific task you need to accomplish. This initial comprehension sets the stage for effective problem-solving.

To truly understand the problem, read it multiple times, identifying key details, constraints, and requirements. Rephrasing the problem in your own words can often clarify its essence. Break down any complex sentences or scenarios into simpler components. This process helps you avoid misinterpretations and ensures you are addressing the problem accurately.

A solid grasp of the problem statement guides your approach. It helps you determine whether permutations, combinations, or other principles are applicable. Without this foundational understanding, attempting to solve the problem can lead to confusion and errors.

Remember, a well-defined problem is halfway solved. Take your time, analyze the information provided, and ensure you're clear on what's being asked. This practice not only enhances your problem-solving skills in combinatorics but also cultivates a mindset that's attentive, analytical, and detail-oriented – qualities that extend beyond mathematics into various facets of problem-solving in the real world.

Visualize the Problem

In the realm of combinatorics, where abstract concepts meet real-world scenarios, visualization becomes an invaluable tool. Many combinatorial problems involve arranging objects, making choices, or following specific sequences. Visualizing the problem allows you to create a mental or graphical representation of the situation, which can provide insights and clarity.

When faced with a combinatorial problem, consider creating diagrams, tables, or even simple models to represent the elements involved. For example, if you're dealing with a permutation problem where you need to arrange people in a queue, you can sketch out a line with placeholders for each person. Visual aids like this help you see patterns, repetitions, or symmetries that might not be immediately apparent from the problem statement alone.

Visualizing the problem doesn't just make it more tangible; it also guides your thought process. As you work on assignments, remember that a clear mental image or a visual representation on paper can lead you to potential solutions more efficiently. Combining mathematical analysis with visual insight allows you to approach problems from multiple angles, increasing your chances of finding an elegant and accurate solution.

Break Down the Problem

Combinatorics problems, especially complex ones, can seem overwhelming at first glance. Breaking down the problem into smaller, manageable components is a powerful strategy to tackle the challenge systematically. This approach not only makes the problem more approachable but also helps you identify patterns and connections that might not be obvious initially.

Start by analyzing the problem statement and identifying its key elements. What are the given values, constraints, and requirements? Then, consider if you can divide the problem into sub-problems or cases. Each sub-problem can be addressed independently, simplifying the overall solution process.

Additionally, breaking down the problem often involves recognizing similarities between different parts of the problem. If you encounter repetitive tasks or scenarios, you can solve one instance and then generalize the solution for others.

By deconstructing the problem, you transform a complex challenge into a series of manageable steps. This approach not only reduces the risk of overlooking crucial details but also boosts your confidence as you steadily work through each component. As you gain experience, you'll become more skilled at identifying how to break down problems effectively, enhancing your problem-solving prowess in combinatorics.

Apply Relevant Concepts

When faced with a combinatorics problem, applying the appropriate concepts is akin to selecting the right tools for a task. Each problem has unique characteristics, and understanding which principles, theorems, or formulas are relevant is crucial. This step requires a deep understanding of the concepts you've studied, so take the time to review and internalize them.

For instance, if the problem involves arranging objects in a specific order, permutations are likely the key. On the other hand, if the order doesn't matter, combinations come into play. Similarly, problems that involve sequential tasks might call for the multiplication principle. By carefully analyzing the problem statement and identifying its underlying structure, you can confidently choose the relevant combinatorial approach.

Double-Check Your Solution

While the sense of accomplishment from finding a solution is undeniable, it's equally important to exercise caution and rigor in your approach. The intricate nature of combinatorial problems leaves room for errors, oversights, or miscalculations. This is where the art of double-checking steps in.

Revisiting your solution with a critical eye helps ensure its accuracy. Verify whether you've accurately applied the chosen concept, whether calculations are error-free, and if your solution aligns with the problem's requirements. Moreover, assessing your solution from different angles can reveal potential shortcuts or alternative methods, leading to a more elegant solution.

The process of double-checking fosters a sense of confidence in your solution and helps you catch any slips that might have otherwise gone unnoticed. Remember, precision is paramount in combinatorics, where small mistakes can lead to vastly different outcomes. By making double-checking an integral part of your problem-solving routine, you're bound to refine your skills and produce consistently reliable solutions.

Practice, Practice, Practice

Practice lies at the heart of mastering combinatorics assignments. As with any skill, consistent practice is essential for honing your problem-solving abilities in this field. Combinatorics problems come in various forms, each presenting unique challenges. By working through a diverse range of problems, you develop a deeper understanding of the underlying concepts and strategies required.

Repetition allows you to internalize the methods used to approach different scenarios. It enhances your ability to recognize patterns, apply relevant principles, and make connections between seemingly unrelated problems. Through practice, you not only improve your technical skills but also gain the confidence to tackle even the most complex combinatorial puzzles.

Remember that it's not just about the quantity of problems you solve, but the quality of your engagement with each one. Analyze your solutions critically, identifying areas for improvement and alternative approaches. With every problem you conquer, you're building a mental toolkit that equips you to face new challenges with resilience and creativity.

Seek Help When Needed

In the realm of combinatorics, seeking help is not a sign of weakness, but a testament to your commitment to understanding and solving problems effectively. When you encounter a particularly challenging or perplexing problem, don't hesitate to reach out for assistance.

Consult your textbooks, online resources, or course materials to gain alternative perspectives on the problem. Sometimes, a fresh explanation can provide the clarity you need to unlock the solution. Engage with peers, teachers, or online communities to discuss the problem. Explaining your thought process to others often leads to insights you might have missed on your own.

Collaboration is a powerful tool in mastering combinatorics. A fellow student might approach a problem from a different angle, offering you a novel way to solve it. By seeking help, you're tapping into a wealth of collective knowledge, fostering a collaborative learning environment, and expanding your problem-solving toolkit.

Remember, the willingness to ask for help demonstrates your dedication to understanding and growing within the field of combinatorics. It's a trait that sets you on a path to becoming a more proficient and confident problem solver.

In conclusion, embarking on the realm of combinatorics assignments is a journey that combines mathematical acumen with strategic thinking. By establishing a strong foundation in fundamental concepts such as permutations, combinations, and the core principles of counting, you lay the groundwork for confidently approaching problems. As you delve into more advanced topics like the inclusion-exclusion principle, generating functions, and Pascal's Triangle, you unlock the tools to tackle even the most intricate combinatorial challenges. Remember, success in combinatorics assignments is not solely about equations and formulas; it's about cultivating a problem-solving mindset. Break down complex problems, visualize scenarios, and apply relevant principles. Mistakes and hurdles are part of the process – learn from them, iterate, and refine your approach.

Combinatorics assignments offer not only mathematical satisfaction but also the opportunity to hone critical thinking and analytical skills that transcend mathematics. So, whether you're arranging objects or counting possibilities, the journey through combinatorics equips you with skills that extend far beyond the classroom, enriching your problem-solving toolkit for a lifetime. Embrace the challenges, persist in your practice, and master the art of counting and arrangement with confidence.

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Title: the rank-ramsey problem and the log-rank conjecture.

Abstract: A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.

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Kodai Senga believes a mechanical issue at least “partially” contributed to his injury this spring, he said, which is why his progress has been paused until he can feel completely comfortable with his delivery.

The Mets ace progressed from no-throw to throwing off flat ground to throwing off mounds to facing hitters, which he did twice last week.

After his pair of live batting practices, though, he told the club that he did not want to continue facing hitters — either in live BPs or with a rehab assignment — until he irons out mechanical problems .

Those mechanical issues are “very technical but to simplify: All my power output was not going towards the catcher,” Senga said Monday through interpreter Hiro Fujiwara. “I wasn’t able to deliver 100 percent of it towards the catcher, which is very important. And when that is happening, I’m more … susceptible to injuries.”

Such as the right shoulder capsule strain that has sidelined him since late February.

Senga said he was playing with a few delivery tweaks this offseason and this spring, “which could have been a factor” in his shoulder problem.

He then felt “some differences” in his delivery during the two live-BP sessions.

“That’s when I thought I should take a little bit of time to re-look at this,” Senga said before the Mets opened a series with the Phillies at Citi Field. “I don’t want to come back into this season and say, ‘Oh, I need a couple more days in-season when I’m not on the IL.’

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Mendoza called Senga “unique” and a “different person.” With others, Mendoza might be more aggressive in trying to push progress forward.

With Senga, Mendoza is trying to listen more than anything else.

“At the end of the day, you don’t want to put your player at risk, especially if he’s not feeling the way he thinks he should be feeling,” the Mets manager said. “We have conversations with some of the other guys that are going through the rehab process, and everybody’s different.

“This is a unique case, situation. And a special player, special talent that’s very meticulous. Every situation is different, and we’ll have to adjust.”

Cultural differences come into play for a player who pitched professionally in Nippon Professional Baseball from 2012-2022.

Senga said that in Japan, rehab steps are “more up to the players,” who are given more autonomy to decide what is best.

“If they feel good, they can keep pushing forward,” Senga said. “Here, the trainers have a very well-structured program.”

Both the Mets and Senga want him back on the mound and healthy .

He last threw a 45-pitch bullpen session Sunday, Mendoza said, and Senga hopes to throw another bullpen session Wednesday.

Senga is not eligible to be activated until May 27, but a return this month is no longer possible. Asked about a timetable, Senga said it was “hard to give you a straight answer.”

“It really depends,” Senga said. “We’ll see how I feel in my next bullpen. If all goes well, it could come sooner than later, but it’s really all up in the air.”

The Mets’ rotation has been solid without Senga, and the rehabbing Tylor Megill and David Peterson offer two more intriguing options, but it lacks a true ace.

Senga was that ace last season and is trying to ensure he can be that ace whenever he returns this season.

“With my current mechanics, I didn’t think I’d be able to come back at 100 percent,” Senga said. “So taking a little bit of time to look over everything, making sure everything is perfect.”

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