Anagrams Of Cogumelo Starting And Ending With 'o' - A Detailed Solution

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Determining the number of anagrams of a word that begin and end with a specific letter is a classic combinatorial problem. In this article, we will delve into the solution of this problem using the word "cogumelo" as an example. We'll break down the steps involved, explain the reasoning behind each step, and provide a clear and concise solution.

Problem statement

How many anagrams of the word "cogumelo" begin and end with the letter "o", considering that there are 1440 possible anagrams?

  • A) 120
  • B) 240
  • C) 480
  • D) 720

Solution

To solve this problem, we need to consider the constraints imposed by the question. The word "cogumelo" has 8 letters, with the letter "o" appearing twice. We want to find the number of anagrams where the first and last letters are both "o".

  1. Fix the first and last letters: Since the anagram must start and end with "o", we can fix these two positions. This leaves us with 6 remaining letters to arrange: "c", "g", "u", "m", "e", and "l".

  2. Calculate the permutations of the remaining letters: We have 6 distinct letters to arrange in the 6 remaining positions. The number of ways to arrange n distinct objects is n! (n factorial), which is the product of all positive integers up to n. In this case, we have 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

Therefore, there are 720 anagrams of the word "cogumelo" that begin and end with the letter "o". The correct answer is D) 720.

Detailed Explanation

Let's delve deeper into the reasoning behind this solution. When we fix the first and last letters as "o", we are essentially reducing the problem to arranging the remaining letters. The key concept here is permutations. A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is given by the formula:

P(n, r) = n! / (n - r)!

In our case, we have 6 distinct letters (c, g, u, m, e, l) and we want to arrange all 6 of them, so n = 6 and r = 6. Therefore, the number of permutations is:

P(6, 6) = 6! / (6 - 6)! = 6! / 0! = 6! / 1 = 720

(Note that 0! is defined as 1).

This confirms our earlier calculation that there are 720 ways to arrange the 6 remaining letters.

Why not consider the two 'o's as distinct?

You might wonder why we didn't initially treat the two "o"s as distinct letters (e.g., "o1" and "o2"). The reason is that the problem asks for anagrams, which are distinct arrangements of letters. If we treated the "o"s as distinct, we would be counting arrangements that are essentially the same (e.g., "o1gumel o2" and "o2gumel o1"). By fixing the "o"s at the beginning and end, we avoid this overcounting.

Alternative approach: considering total anagrams and proportions

While the method described above is the most direct way to solve this problem, we can also consider an alternative approach using the given information that there are 1440 total anagrams of "cogumelo".

  1. Total anagrams: The word "cogumelo" has 8 letters, with the letter "o" appearing twice. The total number of anagrams is given by:

Total anagrams = 8! / 2! = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (2 × 1) = 20160

However, the problem states that there are 1440 possible anagrams. This discrepancy suggests there might be a misunderstanding or a typo in the problem statement regarding the total number of anagrams. Let's proceed with the calculation assuming the given total of 1440 is correct for the sake of exploring this alternative approach.

  1. Proportion of anagrams starting and ending with "o": If we were to randomly pick two positions for the "o"s, there are several possibilities. We are interested in the case where the "o"s are at the beginning and end. Let's think about the probability of this happening.

    • There are 8 positions in the word.
    • We need to choose 2 of these positions for the "o"s.
    • The number of ways to choose 2 positions out of 8 is given by the combination formula:

    C(8, 2) = 8! / (2! × (8 - 2)!) = 8! / (2! × 6!) = (8 × 7) / (2 × 1) = 28

    So, there are 28 possible pairs of positions for the two "o"s.

    Only one of these pairs corresponds to the "o"s being at the beginning and end.

    Therefore, the proportion of anagrams that start and end with "o" is 1/28.

  2. Calculate the number of anagrams: Multiply the total number of anagrams by the proportion:

Number of anagrams starting and ending with "o" = (1/28) × 1440 = 720

This alternative approach, while less direct, arrives at the same answer of 720. It highlights the relationship between combinations, permutations, and probabilities in solving combinatorial problems.

Common mistakes to avoid

  • Forgetting to account for repeated letters: When a word has repeated letters, you need to divide by the factorial of the number of repetitions to avoid overcounting. In this case, we fixed the "o"s, so this wasn't a direct issue in the primary solution, but it's crucial for calculating total anagrams.
  • Confusing permutations and combinations: Permutations consider the order of objects, while combinations do not. In this problem, we are concerned with the order of the letters, so we use permutations.
  • Incorrectly applying the factorial: Make sure you understand the factorial notation (n! = n × (n - 1) × (n - 2) × ... × 1) and how to calculate it correctly.

Conclusion

In conclusion, the number of anagrams of the word "cogumelo" that begin and end with the letter "o" is 720. This problem demonstrates the application of permutation principles in solving combinatorial problems. By carefully considering the constraints and breaking down the problem into smaller steps, we can arrive at the correct solution. Understanding the concepts of permutations, combinations, and factorials is essential for tackling such problems effectively.

This detailed explanation should provide a clear understanding of the solution and the reasoning behind it, making it easier to apply these concepts to other similar problems.

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Tips for Solving Anagram Problems

Anagram problems are a common type of question in mathematics and logic exams. Here are some tips to help you solve them effectively:

  • Identify the constraints: Carefully read the problem statement and identify any constraints, such as the starting and ending letters, or specific letter combinations.
  • Break down the problem: Divide the problem into smaller, manageable steps. For example, fix the constrained letters first and then arrange the remaining letters.
  • Apply the appropriate formulas: Use the permutation or combination formulas as needed. Remember to account for repeated letters by dividing by the factorial of the number of repetitions.
  • Check your answer: If possible, verify your answer using an alternative method or by logical reasoning.
  • Practice regularly: The more you practice anagram problems, the better you will become at solving them quickly and accurately.

By following these tips and understanding the underlying concepts, you can confidently tackle anagram problems in various contexts.

Further Exploration

If you are interested in learning more about anagrams and combinatorial problems, here are some resources you can explore:

  • Textbooks on combinatorics and discrete mathematics
  • Online courses and tutorials on permutations and combinations
  • Websites and forums dedicated to mathematical problem-solving
  • Practice problems and exercises in competitive exam preparation books

By delving deeper into these resources, you can expand your knowledge and skills in this fascinating area of mathematics.

Conclusion

Understanding anagrams and permutations is crucial for solving a variety of mathematical problems, especially those found in competitive exams like the ENEM. The key to mastering these concepts lies in breaking down complex problems into simpler steps and applying the appropriate formulas. In this case, we tackled the specific problem of finding the number of anagrams of the word "cogumelo" that begin and end with the letter "o". We learned that by fixing the positions of the "o"s, we reduced the problem to permuting the remaining letters, resulting in a clear and concise solution. The correct answer, 720, underscores the importance of understanding factorial notation and how to apply it in real-world scenarios. Remember, practice is essential! The more you engage with these types of problems, the more intuitive they will become. Don't be afraid to explore different approaches and learn from your mistakes. With dedication and a solid grasp of the fundamentals, you can confidently tackle any anagram challenge. This exploration extends beyond just finding solutions; it's about developing a critical thinking skillset that can be applied to various aspects of life. So, keep exploring, keep learning, and most importantly, keep challenging yourself!

The word "cogumelo," with its eight letters and a repeating "o," presents a classic example for understanding the concept of anagrams with constraints. Our focus here was not just about finding any arrangement of letters, but specifically those arrangements, or permutations, that begin and end with the letter "o." This constraint adds a layer of complexity, requiring a strategic approach to solve. By fixing the first and last letters as "o," we effectively reduced the problem to arranging the remaining six distinct letters. This is where the concept of permutations of distinct objects comes into play. The formula for permutations, n!, provides the number of ways to arrange n distinct objects, which in our case was 6! (6 factorial). Calculating 6! gives us 720, the number of ways to arrange the letters "c," "g," "u," "m," "e," and "l." This is a direct application of the fundamental principle of counting, a core concept in combinatorics. However, it's crucial to understand why we fixed the "o"s and didn't treat them as distinct letters. If we had, we would have inadvertently counted arrangements that are essentially the same, leading to an incorrect answer. This highlights the importance of carefully interpreting the problem statement and identifying the nuances of the constraints. The alternative approach we discussed, involving the total number of anagrams and proportions, provided another perspective on the problem. While the stated total of 1440 anagrams in the original problem might be inaccurate, exploring this alternative method allowed us to connect the concepts of combinations, permutations, and probabilities. By calculating the probability of the "o"s being at the beginning and end (1/28) and multiplying it by the total number of anagrams, we arrived at the same answer, 720. This demonstrates the interconnectedness of various mathematical concepts and how they can be applied to solve the same problem in different ways. In conclusion, solving anagram problems like this requires a solid understanding of permutations, combinations, and the fundamental principle of counting. By breaking down the problem, identifying the constraints, and applying the appropriate formulas, you can confidently find the solution. Remember, consistent practice and a willingness to explore different approaches are key to mastering these concepts and excelling in problem-solving.

To truly master the art of solving anagram problems, particularly those with constraints like the one presented with the word "cogumelo," it's essential to move beyond simply memorizing formulas and delve into the underlying principles of combinatorial mathematics. Our exploration of the "cogumelo" anagrams highlighted the power of strategic problem-solving. By recognizing that the requirement for the anagram to start and end with the letter "o" significantly simplified the problem, we were able to apply a targeted approach. This involved fixing the positions of the two "o"s, effectively reducing the task to arranging the remaining six distinct letters. This key step showcases the importance of identifying and leveraging constraints in problem-solving. Once the "o"s were fixed, the problem transformed into a classic permutation problem. The concept of permutations deals with the arrangement of objects in a specific order, and the formula n! (n factorial) provides the number of ways to arrange n distinct objects. In our case, n was 6, representing the six remaining letters. Calculating 6! yielded 720, the number of possible arrangements. This direct application of the permutation formula highlights the importance of understanding and correctly applying mathematical tools. However, a crucial aspect of this solution lies in understanding why we treated the two "o"s as indistinguishable after fixing their positions. If we had considered them distinct, we would have overcounted the arrangements, as swapping the two "o"s wouldn't create a new anagram. This subtle point emphasizes the importance of careful consideration of the problem's nuances. The alternative approach we discussed, involving the total number of anagrams and proportions, offered a valuable perspective on the problem. Although the given total of 1440 anagrams might be inaccurate (the actual total is 20160/2 = 10080), exploring this method allowed us to connect permutations with combinations and probabilities. Combinations deal with the selection of objects without regard to order, while probabilities quantify the likelihood of an event occurring. By calculating the probability of the "o"s being at the beginning and end and multiplying it by the (assumed) total number of anagrams, we arrived at the same answer. This demonstrates the interconnectedness of mathematical concepts and the power of approaching problems from different angles. Moreover, this alternative approach highlighted the importance of critical thinking and error detection. The discrepancy between the given total number of anagrams and the calculated total should have raised a red flag, prompting further investigation. This emphasizes the need for double-checking and verifying results in problem-solving. In conclusion, solving anagram problems effectively involves more than just memorizing formulas. It requires a deep understanding of the underlying principles of combinatorial mathematics, strategic problem-solving skills, and the ability to connect different mathematical concepts. By mastering these skills, you can confidently tackle even the most challenging anagram problems and develop a valuable skillset that extends far beyond the realm of mathematics. So, embrace the challenge, explore different approaches, and never stop learning! This is the key to unlocking your mathematical potential and achieving success in problem-solving.