Coefficient Of X⁵y⁵ In (2x-3y)¹⁰: Find It Now!

Hey guys! Let's dive into a fun math problem today. We're going to figure out the coefficient of the $x^5 y^5$ term in the binomial expansion of $(2x - 3y)^{10}$. Sounds like a mouthful, right? Don't worry, we'll break it down step by step so it's super easy to follow. Grab your pencils, and let's get started!

Understanding Binomial Expansion

Before we jump into the problem, let's quickly recap what binomial expansion is all about. The binomial theorem gives us a way to expand expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The general formula looks like this:

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

Here, $\binom{n}{k}$ represents the binomial coefficient, which is also written as "n choose k" and is calculated as:

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

Where $n!$ (n factorial) is the product of all positive integers up to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

The binomial theorem is super useful because it saves us from having to manually multiply out $(a + b)$ by itself $n$ times, especially when $n$ is large. Instead, we can use the formula to directly find the coefficients of each term in the expansion.

In our specific problem, we have $(2x - 3y)^{10}$. So, $a = 2x$, $b = -3y$, and $n = 10$. We're looking for the term with $x^5 y^5$. This means we need to find the value of $k$ such that when we plug it into the formula, we get the desired powers of $x$ and $y$.

Finding the Right Term

Okay, so we need to find the term with $x^5 y^5$ in the expansion of $(2x - 3y)^{10}$. Using the binomial theorem, the general term in the expansion is:

$\binom{10}{k} (2x)^{10-k} (-3y)^k$

We want the power of $x$ to be 5 and the power of $y$ to be 5. This gives us two equations:

$10 - k = 5$ and $k = 5$

Both equations tell us that $k = 5$. This is great because it means there is indeed a term with $x^5 y^5$ in the expansion. Now we just need to plug $k = 5$ into the general term formula and simplify.

So, the term we're interested in is:

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

Now, let's calculate the binomial coefficient and simplify the expression.

Calculating the Coefficient

First, we need to calculate $\binom{10}{5}$:

$\binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$

Now, let's simplify $(2x)^5$ and $(-3y)^5$:

$(2x)^5 = 2^5 x^5 = 32x^5$

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

Now, we plug these values back into the term:

$252 \times 32x^5 \times (-243y^5) = 252 \times 32 \times (-243) x^5 y^5$

Finally, we multiply the numbers together to get the coefficient:

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

So, the coefficient of the $x^5 y^5$ term in the binomial expansion of $(2x - 3y)^{10}$ is -1959552.

Putting It All Together

Alright, let's recap the steps we took to solve this problem:

1. Understand the Binomial Theorem: We started by understanding the binomial theorem and its formula.
2. Identify the Correct Term: We figured out which term in the expansion would have $x^5 y^5$ by setting up equations based on the powers of $x$ and $y$.
3. Calculate the Binomial Coefficient: We calculated $\binom{10}{5}$ using the formula for binomial coefficients.
4. Simplify the Terms: We simplified $(2x)^5$ and $(-3y)^5$.
5. Multiply Everything Together: Finally, we multiplied the binomial coefficient and the simplified terms to get the coefficient of the $x^5 y^5$ term.

By following these steps, we found that the coefficient is -1959552. Not too shabby, huh?

Why This Matters

You might be wondering, "Okay, that's cool, but why do I need to know this?" Well, binomial expansion has applications in various fields, including:

• Probability: Calculating probabilities in scenarios like coin flips or drawing cards.
• Statistics: Approximating complex statistical distributions.
• Computer Science: Analyzing algorithms and data structures.
• Physics: Modeling physical systems.

Understanding binomial expansion can give you a solid foundation for tackling more advanced problems in these areas. Plus, it's just a neat trick to have up your sleeve!

Practice Problems

Want to test your skills? Here are a couple of practice problems you can try:

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

Work through these problems using the steps we outlined above. If you get stuck, don't worry! Just go back and review the material. Practice makes perfect!

Conclusion

So, there you have it! We successfully found the coefficient of the $x^5 y^5$ term in the binomial expansion of $(2x - 3y)^{10}$. Remember, the key is to understand the binomial theorem, identify the correct term, calculate the binomial coefficient, simplify the terms, and multiply everything together. With a little practice, you'll be a binomial expansion pro in no time!

Keep practicing, and don't be afraid to ask questions. Math can be challenging, but it's also super rewarding. Until next time, happy calculating!

Remember: The coefficient of the $x^5 y^5$ term in the expansion of $(2x - 3y)^{10}$ is -1959552. Keep this in mind for your future math adventures! And always double-check your work – even the best mathematicians make mistakes sometimes!