Coefficient Of X^5y^5 In (2x - 3y)^10: A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the fascinating world of binomial expansions, specifically tackling the question: What is the coefficient of the $x^5y^5$ term in the binomial expansion of $(2x - 3y)^{10}$? This might seem daunting at first, but trust me, with a sprinkle of the binomial theorem and a dash of careful calculation, we'll crack this nut in no time. So, buckle up and let's get started!

Understanding the Binomial Theorem

Before we jump into the problem, let's refresh our understanding of the binomial theorem. This theorem is the cornerstone of expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The binomial theorem provides a systematic way to determine the coefficients and terms in the expanded form. It states that:

$$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$$

Where ${n \choose k}$ represents the binomial coefficient, often read as "n choose k," and is calculated as:

$${n \choose k} = \frac{n!}{k!(n-k)!}$$

Here, "!" denotes the factorial, meaning the product of all positive integers up to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120). The binomial coefficient ${n \choose k}$ tells us the number of ways to choose $k$ items from a set of $n$ items, without regard to order. This is crucial for determining the numerical coefficients in our binomial expansion.

Why is this important? Well, the binomial theorem gives us the framework for expanding expressions like $(2x - 3y)^{10}$. It tells us exactly how each term will look, what powers the variables will have, and most importantly, what the coefficients will be. Without this theorem, expanding such an expression would be a tedious and error-prone process. Imagine trying to manually multiply $(2x - 3y)$ by itself ten times! The binomial theorem saves us from that mathematical nightmare.

Let's break down the key components of the binomial theorem:

• (a + b)^n: This is the binomial expression we want to expand. In our case, it's $(2x - 3y)^{10}$.
• ∑ (Summation): This symbol tells us we're adding up a series of terms.
• k = 0 to n: This indicates the range of the summation. We'll be plugging in values for $k$ from 0 up to $n$ (in our case, 10).
• {n \choose k}: This is the binomial coefficient, calculated as n! / (k! * (n-k)!). It determines the numerical coefficient of each term.
• a^(n-k): This is the first term in the binomial (a) raised to the power of (n-k).
• b^k: This is the second term in the binomial (b) raised to the power of k.

So, in essence, the binomial theorem provides a recipe for constructing each term in the expansion. We simply plug in the values for $n$, $k$, $a$, and $b$, and the theorem spits out the corresponding term. We repeat this process for all values of $k$ from 0 to $n$, and then add up all the terms to get the complete expansion.

Identifying the Relevant Term

Now that we've got a solid grasp of the binomial theorem, let's zero in on the specific term we need: the $x^5y^5$ term in the expansion of $(2x - 3y)^{10}$. Our goal is to find the coefficient that sits in front of this term.

Looking back at the binomial theorem formula, we can see that the general term in the expansion is given by:

$${n \choose k} a^{n-k} b^k$$

In our problem:

• $n = 10$ (the exponent of the binomial)
• $a = 2x$ (the first term in the binomial)
• $b = -3y$ (the second term in the binomial)

We want to find the value of $k$ that gives us $x^5y^5$ in the term. This means we need to find $k$ such that:

• $(2x)^{10-k}$ has $x^5$ (so, $10 - k = 5$)
• $(-3y)^k$ has $y^5$ (so, $k = 5$)

Notice that both conditions lead us to the same value: $k = 5$. This is excellent news! It confirms that there is indeed an $x^5y^5$ term in the expansion, and we've found the correct value of $k$ to use in our calculations.

Why is finding the right 'k' so important? Because the value of $k$ dictates everything about the term – the powers of $x$ and $y$, and most importantly, the coefficient. If we used the wrong $k$, we'd end up calculating the coefficient for a completely different term in the expansion. Imagine trying to find the needle in a haystack if you didn't know what a needle looked like! Finding the correct $k$ is like knowing exactly what our needle looks like, making the search much more efficient.

Now that we know $k = 5$, we can plug this value, along with $n = 10$, $a = 2x$, and $b = -3y$, into the general term formula to get the specific term we're interested in:

$${10 \choose 5} (2x)^{10-5} (-3y)^5$$

This is the term that contains $x^5y^5$, and the coefficient we're looking for is the numerical part of this expression. Next, we'll simplify this expression to extract the coefficient.

Calculating the Coefficient

Alright, we're on the home stretch! We've identified the relevant term in the binomial expansion, and now it's time to calculate its coefficient. Recall the term we found:

$${10 \choose 5} (2x)^{10-5} (-3y)^5$$

Let's simplify this step by step:

1. Calculate the binomial coefficient:

   $${10 \choose 5} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$$

2. Simplify the powers:

   $$(2x)^{10-5} = (2x)^5 = 2^5 x^5 = 32x^5$$

   $$(-3y)^5 = (-3)^5 y^5 = -243y^5$$

3. Combine everything:

   $${10 \choose 5} (2x)^5 (-3y)^5 = 252 \times 32x^5 \times (-243)y^5$$

Now, we just need to multiply the numerical coefficients together:

$$252 \times 32 \times (-243) = -1959552$$

Therefore, the coefficient of the $x^5y^5$ term in the expansion of $(2x - 3y)^{10}$ is -1959552. This is a large number, but that's not surprising given the powers involved in the binomial expansion. It's a testament to how quickly these coefficients can grow as the exponent $n$ increases.

Breaking down the calculation: We started by calculating the binomial coefficient ${10 \choose 5}$, which represents the number of ways to choose 5 items from a set of 10. This is a fundamental concept in combinatorics and plays a crucial role in the binomial theorem. Then, we simplified the powers of $(2x)$ and $(-3y)$, remembering to apply the exponent to both the numerical coefficient and the variable. Finally, we multiplied all the numerical coefficients together to arrive at our final answer. Each step is important and builds upon the previous one, highlighting the systematic nature of the binomial theorem.

Identifying the Correct Option

Now that we've calculated the coefficient, let's see which of the given options matches our result. The options were:

A. ${ }_{10} C_5(2)^5(3)^5$ B. ${ }_{10} C_5(2)^5(-3)^5$ C. ${ }_{-10} C _5(2)^5(-3)^5$ D. ${ }_{10} C_5(2)^5(-3)^5$

We can see that option B correctly represents the coefficient we calculated:

$${ }_{10} C_5(2)^5(-3)^5 = 252 \times 32 \times (-243) = -1959552$$

Option A is incorrect because it doesn't include the negative sign from $(-3)^5$. Option C is incorrect because the binomial coefficient ${ }_{-10} C _5$ is not defined in the standard binomial theorem (n must be a non-negative integer). Option D appears to be a duplicate of option B.

Why is it important to match the entire expression, not just the final numerical value? Because the options are designed to test your understanding of the binomial theorem itself, not just your ability to calculate a number. The correct option demonstrates that you understand how the binomial coefficient, the powers of the terms in the binomial, and the signs all come together to form the coefficient of a specific term in the expansion. It's a holistic assessment of your knowledge of the theorem.

Therefore, the correct answer is B. ${ }_{10} C_5(2)^5(-3)^5$. We've successfully navigated the binomial theorem, identified the relevant term, calculated its coefficient, and matched it to the correct option. High five!

Key Takeaways and Further Exploration

Wow, guys, we've covered a lot! Let's recap the key takeaways from our journey through this binomial expansion problem:

• The binomial theorem is your friend: It provides a powerful and systematic way to expand expressions of the form $(a + b)^n$. Master it, and you'll be able to tackle a wide range of expansion problems.
• Identifying the correct term is crucial: Before you start calculating, make sure you've pinpointed the specific term you're interested in. This involves finding the right value of $k$ in the binomial theorem formula.
• Pay attention to signs: The negative sign in $(-3y)$ can make a big difference in the final coefficient. Always be mindful of signs when raising negative numbers to powers.
• Break it down: Complex calculations become much more manageable when you break them down into smaller, more digestible steps.
• Double-check your work: A small error in calculation can lead to a completely wrong answer. Take the time to double-check each step to ensure accuracy.

Now that you've conquered this problem, why not explore the binomial theorem further? Here are some ideas:

• Try different values of n: Expand $(x + y)^n$ for various values of $n$ (e.g., 2, 3, 4, 5) and observe the patterns in the coefficients. This will help you develop a deeper intuition for the binomial theorem.
• Expand binomials with more complex terms: Challenge yourself with expansions like $(x^2 - 2y^3)^8$ or $(1 + x)^{1/2}$ (which leads to the generalized binomial theorem for non-integer exponents).
• Explore applications of the binomial theorem: The binomial theorem has applications in various fields, including probability, statistics, and computer science. Research some of these applications to see the theorem in action.

The beauty of the binomial theorem lies in its versatility and its ability to connect different areas of mathematics. It's a fundamental concept that will serve you well in your mathematical journey. So, keep practicing, keep exploring, and keep expanding your knowledge!

Practice Problems

To solidify your understanding, here are a couple of practice problems similar to the one we just solved:

1. What is the coefficient of the $x^3y^7$ term in the binomial expansion of $(x - 2y)^{10}$?
2. Find the coefficient of the $a^4b^6$ term in the expansion of $(3a + b)^{10}$.

Give these a try, and feel free to share your solutions or any questions you have in the comments below. Happy expanding, everyone!