Finding The Coefficient Of X^2y^3 In (x + Y)^5 A Step-by-Step Solution
In the realm of polynomial expansions, a fascinating question arises: what is the coefficient of the x2y3 term in the expansion of (x + y)^5? This seemingly simple question delves into the heart of the binomial theorem and its applications. To unravel this mathematical puzzle, we embark on a journey of exploration, guided by the principles of combinatorics and algebraic manipulation. This comprehensive guide will not only provide the answer but also illuminate the underlying concepts, empowering you to tackle similar problems with confidence.
Demystifying the Binomial Theorem
The binomial theorem serves as the cornerstone of our exploration, providing a systematic approach to expanding expressions of the form (x + y)^n, where n is a non-negative integer. The theorem elegantly states that:
(x + y)^n = Σ (n choose k) * x^(n-k) * y^k
where the summation (Σ) spans from k = 0 to n, and "(n choose k)" represents the binomial coefficient, also known as the combination formula. The binomial coefficient, denoted as C(n, k) or ⁿCₖ, quantifies the number of ways to choose k objects from a set of n distinct objects, without regard to order. Mathematically, it is defined as:
(n choose k) = n! / (k! * (n - k)!)
where "!" denotes the factorial operation, the product of all positive integers up to the given number. For instance, 5! = 5 * 4 * 3 * 2 * 1 = 120. Understanding the binomial theorem and its components, especially the binomial coefficient, is crucial for deciphering the coefficient of the x2y3 term in our given expression.
Applying the Binomial Theorem to (x + y)^5
To determine the coefficient of x2y3 in the expansion of (x + y)^5, we directly apply the binomial theorem. In this case, n = 5, and we seek the term where the exponent of x is 2 and the exponent of y is 3. This corresponds to the term where k = 3 in the binomial theorem formula. Therefore, the term we are interested in is:
(5 choose 3) * x^(5-3) * y^3 = (5 choose 3) * x^2 * y^3
Now, we need to calculate the binomial coefficient (5 choose 3). Using the combination formula:
(5 choose 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 120 / (6 * 2) = 10
Thus, the term containing x2y3 in the expansion of (x + y)^5 is 10x2y3. This unequivocally reveals that the coefficient of the x2y3 term is 10. This detailed step-by-step application of the binomial theorem underscores its power in efficiently expanding binomial expressions and extracting specific coefficients.
Unveiling the Significance of the Coefficient
The coefficient of x2y3, which we have determined to be 10, holds significant meaning within the context of the polynomial expansion. It represents the number of ways we can obtain the term x2y3 when expanding (x + y)^5. Think of it as selecting 3 'y' terms from the 5 factors of (x + y), and the remaining 2 factors will contribute 'x' terms. The coefficient 10 tells us that there are 10 distinct ways to achieve this specific combination of x^2 and y^3. This combinatorial interpretation adds another layer of understanding to the binomial theorem and its applications. The ability to interpret coefficients in polynomial expansions provides valuable insights into the underlying structure and relationships within mathematical expressions.
Exploring Alternative Approaches
While the binomial theorem provides a direct and efficient method for finding the coefficient of x2y3, it's also beneficial to explore alternative approaches. One such method involves manually expanding (x + y)^5. This approach, though more time-consuming, can provide a deeper understanding of the expansion process. By systematically multiplying (x + y) by itself five times, we can trace how the x2y3 term arises and verify the coefficient. Another approach involves using Pascal's Triangle, a triangular array of numbers where each number is the sum of the two numbers directly above it. The numbers in the 5th row of Pascal's Triangle correspond to the binomial coefficients for the expansion of (x + y)^5. These alternative methods offer valuable perspectives and can serve as a cross-check for the result obtained using the binomial theorem. Understanding multiple approaches to solving a problem enhances mathematical flexibility and problem-solving skills.
Tackling Related Problems
Now that we have successfully determined the coefficient of x2y3 in (x + y)^5, let's extend our understanding by considering related problems. For instance, we can explore finding the coefficient of other terms in the expansion of (x + y)^5, such as the coefficient of x^4y or xy^4. We can also investigate expansions of binomials with different exponents, such as (x + y)^7 or (x + y)^10. Furthermore, we can delve into binomial expansions with more complex terms, such as (2x + 3y)^5 or (x - y)^4. By tackling these related problems, we solidify our grasp of the binomial theorem and its versatility in handling various polynomial expansion scenarios. Each problem solved reinforces the underlying principles and enhances our ability to apply them in diverse contexts.
Conclusion: Mastering the Binomial Theorem
In this comprehensive exploration, we have successfully navigated the realm of polynomial expansions to determine the coefficient of x2y3 in (x + y)^5. We have harnessed the power of the binomial theorem, deciphered the significance of the binomial coefficient, and explored alternative approaches to solving the problem. The journey has not only provided the answer but also illuminated the underlying concepts and techniques. By mastering the binomial theorem and its applications, you equip yourself with a valuable tool for tackling a wide range of mathematical problems, from polynomial expansions to probability calculations. The world of mathematics is vast and interconnected, and the binomial theorem serves as a gateway to further exploration and discovery.
What is the coefficient of the x2y3 term in the polynomial expansion of (x + y)^5?
The question at hand is: In the polynomial expansion of (x + y)^5, what is the coefficient of the x2y3 term? This is a classic problem rooted in the binomial theorem, a fundamental concept in algebra and combinatorics. To solve this, we need to understand how the binomial theorem works and how it can be applied to find specific coefficients in polynomial expansions. This article will provide a detailed explanation of the solution, the underlying principles, and related concepts.
Breaking Down the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. The theorem states:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where:
- Σ represents the summation from k = 0 to n.
- (n choose k) is the binomial coefficient, which is calculated as n! / (k! * (n-k)!).
- n! (n factorial) is the product of all positive integers up to n.
This formula might seem daunting at first, but it's a powerful tool for expanding binomials. Let's break it down further. The binomial coefficient (n choose k) represents the number of ways to choose k items from a set of n items without regard to order. It's also known as a combination. The formula for the binomial coefficient is a cornerstone of combinatorics, providing a way to quantify the number of possible selections. Understanding this coefficient is paramount to grasping the binomial theorem.
Each term in the expansion of (a + b)^n corresponds to a specific value of k in the summation. For each k, we have a term with a binomial coefficient, a power of a, and a power of b. The powers of a and b add up to n, reflecting the degree of the original binomial expression. The binomial theorem provides a systematic way to generate all the terms in the expansion, ensuring that no terms are missed. This systematic approach is what makes the theorem so valuable in algebraic manipulations and problem-solving.
Applying the Binomial Theorem to Our Problem
In our case, we want to find the coefficient of the x2y3 term in the expansion of (x + y)^5. Comparing this to the general form (a + b)^n, we have a = x, b = y, and n = 5. We are looking for the term where the exponent of x is 2 and the exponent of y is 3. To find this term, we need to determine the value of k that satisfies these conditions.
From the binomial theorem, the general term is:
(5 choose k) * x^(5-k) * y^k
We want x^(5-k) to be x^2, so 5 - k = 2, which means k = 3. Let's verify that this also gives us the correct power of y. When k = 3, the power of y is y^3, which matches our requirement. Therefore, the term we are interested in is:
(5 choose 3) * x^(5-3) * y^3 = (5 choose 3) * x^2 * y^3
Now, we need to calculate the binomial coefficient (5 choose 3). Using the formula:
(5 choose 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!)
Let's calculate the factorials:
- 5! = 5 * 4 * 3 * 2 * 1 = 120
- 3! = 3 * 2 * 1 = 6
- 2! = 2 * 1 = 2
Plugging these values into the formula, we get:
(5 choose 3) = 120 / (6 * 2) = 120 / 12 = 10
Therefore, the term containing x2y3 in the expansion of (x + y)^5 is 10x2y3. This definitively tells us that the coefficient of the x2y3 term is 10. This methodical application of the binomial theorem showcases its effectiveness in pinpointing specific terms within polynomial expansions.
The Answer and Its Significance
The coefficient of the x2y3 term in the expansion of (x + y)^5 is 10. This is option D in the given choices. But what does this coefficient actually mean? The coefficient represents the number of ways we can obtain the x2y3 term when expanding (x + y)^5. In other words, it tells us how many different combinations of multiplying terms from the five factors of (x + y) will result in x2y3. This combinatorial interpretation is a key aspect of understanding the binomial theorem and its connection to counting principles. The coefficient not only provides a numerical answer but also offers insights into the underlying structure of the expansion.
Alternative Methods and Verification
While the binomial theorem provides the most efficient method, there are alternative ways to approach this problem, which can serve as a way to verify our answer. One method is to manually expand (x + y)^5, although this can be quite tedious. Another method is to use Pascal's Triangle. Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The rows of Pascal's Triangle correspond to the coefficients in the binomial expansion of (x + y)^n, where n is the row number (starting from 0). The 5th row of Pascal's Triangle is 1, 5, 10, 10, 5, 1, which corresponds to the coefficients in the expansion of (x + y)^5. The term x2y3 corresponds to the fourth term in this sequence (remembering that the powers of y start from 0), which is indeed 10. This verification using Pascal's Triangle reinforces our confidence in the correctness of our solution.
Extending the Concept
The ability to find coefficients in binomial expansions is a valuable skill in mathematics. It has applications in various areas, including probability, statistics, and computer science. For example, if we consider flipping a fair coin 5 times, the probability of getting exactly 3 heads is related to the coefficient of the x^3 term in the expansion of (0.5 + 0.5)^5. By extending our understanding of the binomial theorem, we can tackle more complex problems and appreciate its versatility in different mathematical contexts. The applications of the binomial theorem extend far beyond simple polynomial expansions, highlighting its importance in various fields.
Conclusion: Mastering the Binomial Theorem
In conclusion, we have successfully found the coefficient of the x2y3 term in the expansion of (x + y)^5 using the binomial theorem. The answer is 10, corresponding to option D. We have also discussed the meaning of the coefficient, alternative methods for solving the problem, and the broader applications of the binomial theorem. By mastering the binomial theorem, you gain a powerful tool for solving a wide range of mathematical problems and deepen your understanding of algebraic and combinatorial principles.
