Evaluating Combinations Groups Of Ten From Twelve Items

Lorelei embarks on a mathematical journey to determine the number of distinct groups of ten that can be formed from a set of twelve items. To achieve this, she evaluates the expression $\frac{121}{(12-10) 110!}$, a seemingly complex combinatorial problem that requires careful simplification and calculation. This exploration delves into the intricacies of combinations, factorials, and the fundamental principles that govern these mathematical concepts. Understanding combinations is crucial in various fields, from probability and statistics to computer science and cryptography, making this problem a valuable exercise in mathematical reasoning.

The Initial Expression and Simplification

Lorelei begins with the expression $\frac{121}{(12-10) 110!}$. The initial step involves simplifying the expression within the parentheses, a fundamental operation in mathematical problem-solving. Subtracting 10 from 12 yields 2, transforming the expression into $\frac{121}{2 \cdot 110!}$. This simplification is crucial as it reduces the complexity of the denominator, paving the way for further calculations. Mathematical simplification is a core skill, ensuring that expressions are manageable and easier to manipulate. By addressing the parentheses first, Lorelei demonstrates a clear understanding of the order of operations, a cornerstone of accurate mathematical computation. The simplified expression now presents a clearer picture of the problem, allowing for the subsequent steps to be executed with greater ease. This initial simplification is not just a mechanical step; it's a strategic move that sets the stage for the rest of the solution. The ability to simplify complex expressions is a testament to Lorelei's mathematical acumen and her methodical approach to problem-solving. Moreover, this step highlights the importance of attention to detail, as even a minor error in the initial simplification could cascade through the entire solution, leading to an incorrect answer. In essence, the initial simplification is the bedrock upon which the rest of the solution is built, underscoring its significance in the overall process.

Factorials and the Expansion Process

The expression now involves a factorial, denoted by the exclamation mark (!). A factorial represents the product of all positive integers less than or equal to a given number. For instance, 5! (5 factorial) equals 5 × 4 × 3 × 2 × 1, which is 120. The presence of 110! in the denominator signifies a large number, necessitating careful handling. Lorelei's next step likely involves expanding the factorial or finding a way to simplify it. Understanding factorials is essential in combinatorics, as they often appear in formulas for permutations and combinations. To proceed, Lorelei may need to recognize that 121 can be expressed as a combination of factorials, allowing for cancellation and simplification. The concept of factorials is not merely a mathematical notation; it embodies the essence of counting and arranging items, a fundamental aspect of combinatorics. Expanding factorials is a technique used to reveal the underlying structure of these expressions, making them more amenable to simplification. However, expanding large factorials can be cumbersome, highlighting the need for strategic simplification techniques. Lorelei's ability to navigate the complexities of factorials will be crucial in arriving at the correct solution. The process of expanding factorials is not just about multiplying numbers; it's about understanding the pattern and the underlying mathematical principle that governs these expressions. By carefully expanding the factorial, Lorelei can identify common factors that can be canceled out, significantly reducing the computational burden. This step is a testament to Lorelei's mathematical dexterity and her ability to work with abstract concepts.

Combinations and the Final Calculation

The problem hints at combinations, a way of selecting items from a set without regard to order. The formula for combinations is given by $nCr = \frac{n!}{r!(n-r)!}$, where n represents the total number of items and r represents the number of items being chosen. Combinations are a fundamental concept in combinatorics, allowing us to calculate the number of ways to select a subset from a larger set. In this case, Lorelei aims to determine the number of ways to choose 10 items from 12, which can be represented as 12C10. Understanding the combination formula is paramount to solving this problem. Lorelei likely needs to connect the given expression to this formula to arrive at the correct answer. The final calculation will involve substituting the appropriate values into the combination formula and simplifying the resulting expression. This step requires a thorough understanding of the relationship between factorials and combinations, as well as the ability to perform arithmetic operations accurately. Calculating combinations is not just a matter of applying a formula; it's about understanding the underlying principle of selection and arrangement. Lorelei's final calculation will demonstrate her mastery of these concepts and her ability to apply them in a practical context. The accurate calculation of combinations is crucial in various fields, from probability and statistics to cryptography and computer science. Lorelei's ability to perform this calculation flawlessly will underscore her mathematical proficiency and her attention to detail.

Analysis of Lorelei's Solution (Incomplete)

Lorelei's provided solution is incomplete, with the second step presented as "2. Expand: $\frac{6 \cdot 5 \cdotDiscussion category :mathematics" This suggests she intended to expand a factorial or a similar expression, but the expansion is not fully shown. Analyzing incomplete solutions is a crucial part of mathematical problem-solving, allowing us to identify potential errors or omissions. To complete the solution, we need to understand what Lorelei was trying to expand and why. The presence of "6" and "5" in the expansion hints at a factorial or a product of consecutive integers. Identifying the pattern in the expansion is key to reconstructing the complete solution. It's possible that Lorelei was trying to calculate 12C10 directly or simplify the expression obtained in the first step. To evaluate her solution, we need to compare her approach with the standard method for calculating combinations. Comparing different solution methods is a valuable exercise in mathematical problem-solving, allowing us to identify the most efficient and accurate approach. Without the complete expansion, it's difficult to determine whether Lorelei's solution is correct or not. Further steps would involve completing the expansion, simplifying the expression, and comparing the result with the actual value of 12C10. This analysis underscores the importance of showing all the steps in a mathematical solution, as it allows for easier verification and identification of potential errors.

Completing the Solution and Finding 12C10

To determine the number of groups of ten that can be made out of twelve items, we need to calculate 12C10, which is equivalent to $\frac{12!}{10!(12-10)!}$. Calculating 12C10 is the ultimate goal of this problem, and it requires a systematic application of the combination formula. This expression can be simplified to $\frac{12!}{10!2!}$. Simplifying the expression is crucial to making the calculation manageable. Expanding the factorials, we get $\frac{12 \cdot 11 \cdot 10!}{10! \cdot 2 \cdot 1}$. Expanding the factorials strategically allows us to cancel out common terms, simplifying the expression further. The 10! in the numerator and denominator cancel out, leaving us with $\frac{12 \cdot 11}{2 \cdot 1}$. Canceling out common terms is a fundamental simplification technique in mathematics. This simplifies to $\frac{132}{2}$, which equals 66. Performing the final calculation yields the answer, representing the number of ways to choose 10 items from 12. Therefore, there are 66 different groups of ten that Lorelei can make out of twelve items. Interpreting the result in the context of the problem is the final step, ensuring that the answer makes sense and addresses the original question. This complete solution demonstrates the process of calculating combinations, highlighting the importance of simplification, factorial manipulation, and the application of the combination formula.

Lorelei's attempt to evaluate the expression $\frac{121}{(12-10) 110!}$ to determine the number of groups of ten from twelve items is a valuable exercise in combinatorics. While her solution is incomplete, the initial steps demonstrate an understanding of simplification and factorials. By completing the calculation using the combination formula, we find that there are 66 different groups of ten that can be formed. This exploration highlights the importance of understanding combinations, factorials, and the systematic application of mathematical principles to solve real-world problems. Mathematical problem-solving is not just about arriving at the correct answer; it's about understanding the underlying concepts and the process of arriving at the solution. Lorelei's journey through this problem underscores the importance of persistence, attention to detail, and a solid foundation in mathematical principles.