Calculating Combinations Lorelei's Approach To Grouping Items

by Viktoria Ivanova 62 views

Let's dive into a fascinating mathematical problem Lorelei tackled, exploring the concept of combinations and how to calculate them. She aimed to find out how many different groups of ten items she could create from a pool of twelve distinct items. The expression she worked with was 12!(12−10)!10!\frac{12!}{(12-10)!10!}, which represents the number of combinations of choosing 10 items from 12, often written as ¹²C₁₀ or (¹²₁₀). Understanding this problem requires us to grasp the fundamentals of factorials and combinations, so let's break it down step by step.

Understanding Factorials and Combinations

Factorials are the backbone of this calculation. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials help us count the number of ways to arrange items in a specific order. However, when the order doesn't matter, like in Lorelei's problem, we turn to combinations.

Combinations, on the other hand, focus on selecting groups of items where the order of selection is irrelevant. The formula for combinations, which Lorelei's expression embodies, is:

nCr = n! / (r! * (n-r)!)

Where:

  • n is the total number of items.
  • r is the number of items to choose.
  • nCr represents the number of combinations of choosing r items from n.

In Lorelei's case, n = 12 (total items) and r = 10 (items to choose). So, we're looking at ¹²C₁₀. This means we want to find out how many different groups of 10 items can be formed from a set of 12 items, without considering the order in which they are chosen.

Lorelei's Initial Steps and Where Things Might Have Gone Astray

Lorelei's initial expression was 12!(12−10)!10!\frac{12!}{(12-10)!10!}. Let's analyze her steps:

  1. Subtract within parentheses and simplify: 61(2)∣51\frac{61}{(2) \mid 51}

    This step appears to have a few issues. First, the subtraction within the parentheses (12-10) should result in 2, which is correct in her expression. However, the numbers 61, 51, and the vertical bar "|" are completely out of context and seem to be a mistake. It's unclear where these numbers originated, and they don't logically follow from the original expression. The factorial notation (!) seems to have been misinterpreted or omitted in this step.

  2. Expanel: $\frac{6 \cdot 5 Discussion category : mathematics

    This step is also problematic. The term "Expanel" is not a standard mathematical term, and the expression $\frac{6 \cdot 5 Discussion category : mathematics" does not logically follow from the previous incorrect step. It seems Lorelei might have been trying to simplify the factorials, but the execution is flawed. The inclusion of "Discussion category : mathematics" at the end is also perplexing and irrelevant to the calculation.

It's clear that Lorelei's solution has gone off track early on. Let's correctly solve the problem to understand the proper approach.

Correctly Evaluating the Expression

To correctly evaluate 12!(12−10)!10!\frac{12!}{(12-10)!10!}, we need to follow the order of operations and understand how to simplify factorials within combinations.

  1. Simplify within the parentheses:

    (12 - 10) = 2

    So, the expression becomes 12!2!10!\frac{12!}{2!10!}.

  2. Expand the factorials:

    12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

    2! = 2 × 1 = 2

    10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

    Now we have 12×11×10×9×8×7×6×5×4×3×2×1(2×1)(10×9×8×7×6×5×4×3×2×1)\frac{12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{(2 × 1)(10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)}.

  3. Simplify by canceling out common factors:

    Notice that 10! appears in both the numerator and the denominator. We can cancel it out:

    12×11×10!2!10!=12×112×1\frac{12 × 11 × 10!}{2!10!} = \frac{12 × 11}{2 × 1}

  4. Calculate the result:

    12×112=1322=66\frac{12 × 11}{2} = \frac{132}{2} = 66

Therefore, ¹²C₁₀ = 66. This means there are 66 different groups of ten items that Lorelei can make out of twelve items.

Common Mistakes and How to Avoid Them

Lorelei's attempt highlights several common mistakes when dealing with combinations and factorials:

  • Misinterpreting Factorial Notation: It's crucial to understand that n! represents the product of all positive integers up to n. Avoid simply writing down random numbers.
  • Incorrect Simplification: When simplifying expressions with factorials, look for opportunities to cancel out common factors. This significantly reduces the amount of calculation needed.
  • Order of Operations: Always follow the correct order of operations (PEMDAS/BODMAS). Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
  • Understanding the Formula: Make sure you correctly understand and apply the combination formula (nCr = n! / (r! * (n-r)!)).

To avoid these mistakes, practice simplifying factorial expressions and working through combination problems. Double-check your calculations and ensure each step logically follows from the previous one.

The Significance of Combinations

Combinations are a fundamental concept in mathematics and have wide-ranging applications in various fields, including:

  • Probability: Combinations are used to calculate the probability of events, such as the odds of winning a lottery or drawing specific cards from a deck.
  • Statistics: Combinations are essential in statistical analysis, particularly in sampling and experimental design.
  • Computer Science: Combinatorial algorithms are used in various applications, such as data mining, cryptography, and network optimization.
  • Real-World Scenarios: Combinations help us solve everyday problems, such as determining the number of ways to form a committee from a group of people or selecting a team from a roster of players.

By understanding combinations, we gain valuable tools for analyzing and solving problems involving selection and grouping.

Conclusion

Lorelei's journey to evaluate 12!(12−10)!10!\frac{12!}{(12-10)!10!} serves as a valuable lesson in the importance of understanding fundamental mathematical concepts and applying them correctly. While her initial attempt contained errors, by breaking down the problem, understanding factorials and combinations, and following the correct steps, we were able to arrive at the correct solution: 66. Remember, practice makes perfect, and a solid grasp of mathematical principles opens doors to solving a wide array of problems in various fields. So, keep exploring, keep learning, and keep those combinatorial calculations coming!

By mastering these concepts, you'll be well-equipped to tackle similar problems and appreciate the power and elegance of mathematics in solving real-world challenges. Keep practicing, keep exploring, and you'll become a combinatorial whiz in no time!