Coefficient Of X⁵y⁵ In (2x - 3y)¹⁰ A Binomial Theorem Exploration

Jul 10, 2025 by ADMIN 66 views

Unveiling the Coefficient of the x⁵y⁵ Term in the Binomial Expansion of (2x - 3y)¹⁰

In the realm of algebra, binomial expansion stands as a fundamental concept, offering a systematic approach to expanding expressions of the form (a + b)ⁿ. This article delves into the intricacies of binomial expansion, specifically focusing on the task of identifying the coefficient of the x⁵y⁵ term within the expansion of (2x - 3y)¹⁰. We will embark on a comprehensive exploration, dissecting the binomial theorem, unraveling the steps involved in calculating the coefficient, and shedding light on the underlying principles that govern this mathematical phenomenon.

The Binomial Theorem: A Gateway to Expansion

At the heart of binomial expansion lies the binomial theorem, a cornerstone of algebraic manipulation. This theorem provides a precise formula for expanding expressions of the form (a + b)ⁿ, where 'a' and 'b' represent any algebraic terms and 'n' is a non-negative integer. The theorem elegantly expresses the expansion as a sum of terms, each involving a binomial coefficient, powers of 'a', and powers of 'b'.

The binomial theorem states that:

(a + b)ⁿ = Σ (ⁿCᵣ) a^(n-r) b^r

where:

Σ denotes the summation symbol, indicating the sum of terms.
ⁿCᵣ represents the binomial coefficient, often read as "n choose r," which calculates the number of ways to choose 'r' objects from a set of 'n' objects. It is mathematically defined as n! / (r! (n-r)!).
a^(n-r) represents 'a' raised to the power of (n-r).
b^r represents 'b' raised to the power of 'r'.
r is an index that ranges from 0 to n, encompassing all terms in the expansion.

The binomial theorem serves as our guiding light, providing the framework for unraveling the expansion of (2x - 3y)¹⁰ and pinpointing the coefficient of the x⁵y⁵ term.

Dissecting the Expansion of (2x - 3y)¹⁰

Now, let's apply the binomial theorem to the specific expression (2x - 3y)¹⁰. Our goal is to determine the coefficient of the term containing x⁵y⁵. To achieve this, we need to identify the term in the expansion where the power of 'x' is 5 and the power of 'y' is also 5. This requires a careful examination of the binomial theorem formula and how it applies to our expression.

Comparing (2x - 3y)¹⁰ with the general form (a + b)ⁿ, we can make the following correspondences:

a = 2x
b = -3y (Note the negative sign, which is crucial)
n = 10

Substituting these values into the binomial theorem formula, we get:

(2x - 3y)¹⁰ = Σ (¹⁰Cᵣ) (2x)^(10-r) (-3y)^r

Our quest now narrows down to finding the value of 'r' that results in the term x⁵y⁵. This occurs when the exponent of 'x' (10 - r) is equal to 5 and the exponent of 'y' (r) is also equal to 5. Solving the equation 10 - r = 5, we find that r = 5. This confirms that the term with r = 5 will indeed contain x⁵y⁵.

Unveiling the Coefficient: A Step-by-Step Calculation

With the value of 'r' determined, we can now focus on calculating the coefficient of the x⁵y⁵ term. We substitute r = 5 into the general term formula:

(¹⁰C₅) (2x)^(10-5) (-3y)⁵

This simplifies to:

(¹⁰C₅) (2x)⁵ (-3y)⁵

Now, let's break down the calculation into smaller steps:

Calculate the binomial coefficient ¹⁰C₅:

¹⁰C₅ = 10! / (5! 5!) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1) = 252
Calculate (2x)⁵:

(2x)⁵ = 2⁵ x⁵ = 32x⁵
Calculate (-3y)⁵:

(-3y)⁵ = (-3)⁵ y⁵ = -243y⁵
Combine the results:

(¹⁰C₅) (2x)⁵ (-3y)⁵ = 252 × 32x⁵ × (-243y⁵) = -1959552 x⁵y⁵

Therefore, the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰ is -1959552.

The Final Verdict: Deciphering the Answer Choices

Having meticulously calculated the coefficient, let's examine the provided answer choices:

A. ¹⁰C₅ (2)⁵ (3)⁵ = 252 * 32 * 243 = 1959552 B. ¹⁰C₅ (2)⁵ (3) = 252 * 32 * 3 = 24192 C. ¹⁰C₅ (2)⁵ (-3)⁵ = 252 * 32 * (-243) = -1959552 D. ¹⁰C₅ = 252

Comparing our calculated coefficient (-1959552) with the answer choices, we find that option C accurately represents the coefficient. Option A is close but misses the crucial negative sign, while options B and D are significantly different from the correct value.

Key Takeaways and Conclusion

In this exploration, we have successfully navigated the realm of binomial expansion, focusing on the specific task of determining the coefficient of the x⁵y⁵ term in the expansion of (2x - 3y)¹⁰. We have witnessed the power of the binomial theorem, a fundamental tool in algebraic manipulation, and its ability to systematically expand expressions of the form (a + b)ⁿ.

Our journey involved several key steps:

Understanding the binomial theorem and its formula.
Applying the theorem to the given expression, (2x - 3y)¹⁰.
Identifying the term containing x⁵y⁵ by determining the appropriate value of 'r'.
Calculating the binomial coefficient and the powers of the terms.
Combining the results to arrive at the coefficient of the x⁵y⁵ term.
Comparing the calculated coefficient with the provided answer choices to identify the correct solution.

Through this process, we have not only pinpointed the coefficient but also gained a deeper appreciation for the elegance and precision of the binomial theorem. Binomial expansion is a cornerstone of algebra, with applications extending to various fields, including probability, statistics, and calculus. Mastering this concept opens doors to a broader understanding of mathematical relationships and problem-solving techniques.

Introduction: The Binomial Theorem and Its Power

In the captivating world of mathematics, the binomial theorem stands as a beacon of algebraic prowess. It elegantly unravels the expansion of expressions in the form (a + b)ⁿ, where 'n' is a non-negative integer. This theorem has far-reaching applications, from simplifying complex equations to laying the groundwork for probability calculations. In this comprehensive guide, we embark on a journey to decipher the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰. Our exploration will encompass the intricacies of the binomial theorem, its application to the given expression, and a step-by-step calculation to unveil the coveted coefficient. Understanding the binomial theorem and mastering its applications is a critical skill in algebra and beyond.

Unveiling the Binomial Theorem: The Foundation of Expansion

The cornerstone of our endeavor is the binomial theorem itself. It provides a precise formula for expanding expressions of the form (a + b)ⁿ, expressing the expansion as a sum of terms, each a product of a binomial coefficient, a power of 'a', and a power of 'b'. This theorem is not just a formula; it's a powerful tool that simplifies complex algebraic manipulations. The binomial theorem states:

(a + b)ⁿ = Σ (ⁿCᵣ) a^(n-r) b^r

Let's break down the components of this theorem:

Σ: This symbol represents summation, indicating that we're adding up a series of terms.
ⁿCᵣ: This is the binomial coefficient, read as "n choose r," which signifies the number of ways to select 'r' objects from a set of 'n' distinct objects. It's calculated as n! / (r! (n-r)!). This coefficient is crucial for determining the numerical factor in each term of the expansion.
a^(n-r): This represents 'a' raised to the power of (n-r).
b^r: This represents 'b' raised to the power of 'r'.
r: This index varies from 0 to n, ensuring we capture all terms in the expansion.

This formula is the key to unlocking the expansion of binomial expressions, allowing us to systematically determine each term and its corresponding coefficient. Understanding each component of the binomial theorem is essential for accurate calculations.

Applying the Binomial Theorem to (2x - 3y)¹⁰: Setting the Stage

Now, let's put the binomial theorem into action by applying it to our specific expression, (2x - 3y)¹⁰. Our objective is to pinpoint the coefficient of the term containing x⁵y⁵. This requires careful substitution and analysis of the binomial theorem formula. This is where the practical application of the theorem begins, and precision is key.

By comparing (2x - 3y)¹⁰ with the general form (a + b)ⁿ, we can establish the following correspondences:

a = 2x
b = -3y (Note the critical negative sign! This is a common mistake to overlook.)
n = 10

Substituting these values into the binomial theorem formula, we obtain:

(2x - 3y)¹⁰ = Σ (¹⁰Cᵣ) (2x)^(10-r) (-3y)^r

Our next task is to identify the value of 'r' that yields the term x⁵y⁵. This means finding the 'r' such that the exponent of 'x' (10 - r) is 5 and the exponent of 'y' (r) is also 5. This step requires logical deduction and a clear understanding of how exponents work within the binomial theorem.

Identifying the Target Term: Finding the Right 'r' Value

The core of our problem lies in finding the term that contains x⁵y⁵. This is where we need to strategically use the information we have to determine the correct value of 'r'. To do this, we focus on the exponents of 'x' and 'y' in the general term of the expansion. The exponent of 'x' is (10 - r), and the exponent of 'y' is 'r'. Careful observation and algebraic manipulation are needed here.

We want the exponent of 'x' to be 5, so we set up the equation:

10 - r = 5

Solving for 'r', we get:

r = 5

This elegantly confirms that when r = 5, the term will indeed contain x⁵. Furthermore, since the exponent of 'y' is 'r', and r = 5, the term will also contain y⁵. Therefore, we have successfully identified the term we need to analyze. This step highlights the power of algebraic manipulation in solving mathematical problems.

Calculating the Coefficient: A Step-by-Step Approach

With the value of 'r' firmly established, we can now embark on the calculation of the coefficient for the x⁵y⁵ term. We'll methodically substitute r = 5 into the general term formula and simplify. This requires a systematic approach to avoid errors and ensure accuracy.

Substituting r = 5 into the general term formula, we have:

(¹⁰C₅) (2x)^(10-5) (-3y)⁵

Simplifying the exponents, we get:

(¹⁰C₅) (2x)⁵ (-3y)⁵

Now, let's break down the calculation into smaller, manageable steps:

Calculate the binomial coefficient ¹⁰C₅:

¹⁰C₅ = 10! / (5! 5!) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1) = 252

This step involves calculating a factorial and simplifying the fraction. Attention to detail is crucial here.
Calculate (2x)⁵:

(2x)⁵ = 2⁵ x⁵ = 32x⁵

Remember to apply the exponent to both the constant and the variable. This is a fundamental rule of exponents.
Calculate (-3y)⁵:

(-3y)⁵ = (-3)⁵ y⁵ = -243y⁵

Pay close attention to the negative sign! Raising a negative number to an odd power results in a negative number. This is a common source of errors if overlooked.
Combine the results:

(¹⁰C₅) (2x)⁵ (-3y)⁵ = 252 × 32x⁵ × (-243y⁵) = -1959552 x⁵y⁵

Finally, we multiply all the calculated components together. This step requires careful arithmetic.

Thus, the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰ is -1959552. This meticulous calculation showcases the power of step-by-step problem-solving.

Identifying the Correct Answer: A Moment of Truth

With the coefficient calculated, we arrive at the final stage: comparing our result with the provided answer choices. This is a crucial step to ensure we have not made any errors and to confidently select the correct answer. Verification is an essential part of the problem-solving process.

Let's revisit the answer choices:

A. ¹⁰C₅ (2)⁵ (3)⁵ = 252 * 32 * 243 = 1959552 B. ¹⁰C₅ (2)⁵ (3) = 252 * 32 * 3 = 24192 C. ¹⁰C₅ (2)⁵ (-3)⁵ = 252 * 32 * (-243) = -1959552 D. ¹⁰C₅ = 252

By carefully comparing our calculated coefficient (-1959552) with the answer choices, we can confidently identify option C as the correct answer. Option A is very close but misses the crucial negative sign, which highlights the importance of paying attention to details. The other options are significantly different, reinforcing our confidence in our calculation. This final comparison solidifies our understanding and demonstrates the importance of accuracy in mathematical problem-solving.

Conclusion: Mastering the Binomial Theorem

In this comprehensive exploration, we have successfully navigated the intricacies of binomial expansion and determined the coefficient of the x⁵y⁵ term in the expansion of (2x - 3y)¹⁰. We have not only found the answer but also deepened our understanding of the binomial theorem and its applications. Mastering this theorem opens doors to a broader range of algebraic problems and lays a solid foundation for more advanced mathematical concepts.

Our journey has involved several key takeaways:

The binomial theorem provides a systematic way to expand expressions of the form (a + b)ⁿ.
The binomial coefficient (ⁿCᵣ) plays a crucial role in determining the numerical factors in the expansion.
Careful attention to signs, especially negative signs, is essential for accurate calculations.
Breaking down complex problems into smaller, manageable steps simplifies the process and reduces errors.
Verification and comparison with answer choices are crucial for ensuring accuracy.

By following these principles and practicing the application of the binomial theorem, you can confidently tackle a wide range of algebraic problems and unlock the beauty and power of mathematics. This journey has demonstrated that with persistence, attention to detail, and a solid understanding of fundamental concepts, even complex mathematical problems can be solved with clarity and precision.

Introduction: Demystifying Binomial Expansion

Binomial expansion is a fundamental concept in algebra, providing a systematic method for expanding expressions of the form (a + b)ⁿ. This article serves as a comprehensive guide to demystify this concept, focusing on the specific problem of finding the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰. We will delve into the binomial theorem, outline the step-by-step process for calculating the coefficient, and emphasize the importance of precision in algebraic manipulation. Understanding binomial expansion is a crucial skill for success in algebra and related fields.

The Binomial Theorem: A Powerful Tool for Expansion

At the heart of binomial expansion lies the binomial theorem, a powerful formula that provides a structured approach to expanding expressions of the form (a + b)ⁿ, where 'n' is a non-negative integer. This theorem expresses the expansion as a sum of terms, each involving a binomial coefficient, a power of 'a', and a power of 'b'. The binomial theorem is not just a formula; it's a roadmap for expanding complex expressions. A solid understanding of this theorem is essential for tackling binomial expansion problems.

The binomial theorem states:

(a + b)ⁿ = Σ (ⁿCᵣ) a^(n-r) b^r

Let's dissect this formula to understand its components:

Σ: This symbol represents summation, indicating that we are summing a series of terms.
ⁿCᵣ: This is the binomial coefficient, pronounced "n choose r," representing the number of ways to select 'r' objects from a set of 'n' distinct objects. It is calculated as n! / (r! (n-r)!). The binomial coefficient is a key factor in determining the magnitude of each term in the expansion.
a^(n-r): This term represents 'a' raised to the power of (n-r).
b^r: This term represents 'b' raised to the power of 'r'.
r: This is the index of summation, ranging from 0 to n, ensuring that we account for all terms in the expansion.

With this formula in hand, we have the necessary tools to tackle our problem. The binomial theorem is the foundation upon which our solution will be built.

Applying the Theorem to (2x - 3y)¹⁰: Setting Up the Problem

To apply the binomial theorem to our specific problem, we need to substitute the appropriate values into the formula. Our goal is to find the coefficient of the x⁵y⁵ term in the expansion of (2x - 3y)¹⁰. This requires careful substitution and a clear understanding of how the theorem applies to our specific expression. Precision is paramount at this stage.

Comparing (2x - 3y)¹⁰ with the general form (a + b)ⁿ, we can identify the following correspondences:

a = 2x
b = -3y (Crucially, note the negative sign. This is a frequent point of error.)
n = 10

Substituting these values into the binomial theorem formula, we get:

(2x - 3y)¹⁰ = Σ (¹⁰Cᵣ) (2x)^(10-r) (-3y)^r

Our next step is to determine the value of 'r' that corresponds to the x⁵y⁵ term. This involves analyzing the exponents of 'x' and 'y' in the general term of the expansion. Logical reasoning is required to connect the general form to the specific term we are seeking.

Pinpointing the Target Term: Finding the Value of 'r'

The heart of our problem lies in identifying the term containing x⁵y⁵. To achieve this, we need to determine the value of 'r' that results in the exponents of 'x' and 'y' being 5. This requires a careful examination of the general term in the binomial expansion. Strategic thinking will guide us to the solution.

We focus on the exponents of 'x' and 'y' in the general term:

Exponent of x: (10 - r)
Exponent of y: r

We want the exponent of 'x' to be 5, so we set up the equation:

10 - r = 5

Solving for 'r', we find:

r = 5

This confirms that when r = 5, the term will contain x⁵. Since the exponent of 'y' is also 'r', the term will also contain y⁵. We have successfully identified the value of 'r' that corresponds to the term we are interested in. This step demonstrates the power of algebraic problem-solving.

Calculating the Coefficient: A Step-by-Step Breakdown

Now that we have determined the value of 'r', we can proceed with calculating the coefficient of the x⁵y⁵ term. We substitute r = 5 into the general term formula and simplify the expression. This process requires a systematic approach and careful attention to detail.

Substituting r = 5 into the general term formula, we obtain:

(¹⁰C₅) (2x)^(10-5) (-3y)⁵

Simplifying the exponents, we have:

(¹⁰C₅) (2x)⁵ (-3y)⁵

To make the calculation more manageable, we break it down into individual steps:

Calculate the binomial coefficient ¹⁰C₅:

¹⁰C₅ = 10! / (5! 5!) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1) = 252

This step involves calculating factorials and simplifying the resulting fraction. Accuracy is crucial in this calculation.
Calculate (2x)⁵:

(2x)⁵ = 2⁵ x⁵ = 32x⁵

Remember to apply the exponent to both the coefficient and the variable. This is a fundamental rule of exponents.
Calculate (-3y)⁵:

(-3y)⁵ = (-3)⁵ y⁵ = -243y⁵

Pay close attention to the negative sign! Raising a negative number to an odd power results in a negative number. This is a common source of error if overlooked.
Combine the results:

(¹⁰C₅) (2x)⁵ (-3y)⁵ = 252 × 32x⁵ × (-243y⁵) = -1959552 x⁵y⁵

Finally, we multiply all the calculated components together to obtain the coefficient. This step requires careful arithmetic.

Therefore, the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰ is -1959552. This methodical calculation highlights the importance of breaking down complex problems into simpler steps.

Verifying the Solution: Choosing the Correct Answer

Having calculated the coefficient, we now need to compare our result with the provided answer choices to ensure accuracy. This step is crucial for verifying our solution and building confidence in our answer. Verification is an essential part of the problem-solving process.

Let's review the answer choices:

A. ¹⁰C₅ (2)⁵ (3)⁵ = 252 * 32 * 243 = 1959552 B. ¹⁰C₅ (2)⁵ (3) = 252 * 32 * 3 = 24192 C. ¹⁰C₅ (2)⁵ (-3)⁵ = 252 * 32 * (-243) = -1959552 D. ¹⁰C₅ = 252

By comparing our calculated coefficient (-1959552) with the answer choices, we can confidently select option C as the correct answer. Option A is similar but lacks the crucial negative sign, emphasizing the importance of attention to detail. The other options are significantly different, further solidifying our confidence in our solution. This final comparison demonstrates the value of checking your work and ensuring accuracy.

Conclusion: Mastering Binomial Expansion

In this comprehensive guide, we have successfully navigated the process of finding the coefficient of the x⁵y⁵ term in the binomial expansion of (2x - 3y)¹⁰. We have explored the binomial theorem, outlined the step-by-step calculation process, and emphasized the importance of accuracy and precision in algebraic manipulation. Mastering binomial expansion empowers you to tackle a wide range of algebraic problems with confidence.

Key takeaways from our journey include:

The binomial theorem provides a systematic method for expanding expressions of the form (a + b)ⁿ.
The binomial coefficient (ⁿCᵣ) is a crucial component in determining the terms of the expansion.
Careful attention to signs, particularly negative signs, is essential for accurate calculations.
Breaking down complex problems into manageable steps simplifies the process and reduces errors.
Verifying your solution against provided answer choices ensures accuracy and builds confidence.

By internalizing these principles and practicing the application of the binomial theorem, you can unlock the power of algebraic manipulation and achieve success in mathematics. This detailed exploration has highlighted that with systematic problem-solving and a strong grasp of fundamental concepts, you can conquer even complex mathematical challenges. Remember, practice and attention to detail are key to mastering binomial expansion and other algebraic concepts.