Binomial Expansion: Finding The Sixth Term Of (2a-3b)^10

Alright, guys, let's dive into a binomial expansion problem! Specifically, we need to figure out which expression correctly represents the sixth term in the expansion of $(2a - 3b)^{10}$. This might sound intimidating, but don't worry, we'll break it down step by step. Understanding binomial expansion is super useful in various fields like probability, statistics, and even engineering, so let’s get started!

Understanding the Binomial Theorem

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

$(x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k$

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

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

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 essentially tells us how to expand a binomial expression raised to a power, and it's the foundation for solving our problem. Remember that each term in the expansion follows a specific pattern, which we'll use to find the sixth term.

Identifying the Correct Term

In our case, we have $(2a - 3b)^{10}$, and we want to find the sixth term. Notice that in the binomial theorem formula, the term number is related to the index $k$. Specifically, the first term corresponds to $k = 0$, the second term corresponds to $k = 1$, and so on. Therefore, the sixth term will correspond to $k = 5$.

So, we can plug in the values into the binomial theorem formula. Here, $x = 2a$, $y = -3b$, and $n = 10$. We want the term when $k = 5$:

${10 \choose 5} (2a)^{10-5} (-3b)^5$

Now, let's simplify this expression:

${10 \choose 5} (2a)^5 (-3b)^5$

We can calculate the binomial coefficient ${10 \choose 5}$ as:

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

So, the sixth term is:

$252 (2a)^5 (-3b)^5$

Now, let's look at the given options and see which one matches our result.

Analyzing the Options

We need to find the option that matches ${10 \choose 5} (2a)^5 (-3b)^5$:

A. ${ }_{10} C_6(-2 a)^4(3 b)^6$ B. ${ }_{10} C_5(-2 a)^5(3 b)^5$ C. ${ }_{10} C_6(2 a)^6(-3 b)^4$ D. ${ }_{10} C_6(2 a)^4(-3 b)^6$ E. ${ }_{10} C_5(2 a)^5(-3 b)^5$

Let's break down each option:

• Option A: ${ }_{10} C_6(-2 a)^4(3 b)^6$ - This is incorrect because it uses $k=6$ and has incorrect powers and signs.
• Option B: ${ }_{10} C_5(-2 a)^5(3 b)^5$ - This is incorrect because it has $(-2a)$ instead of $(2a)$ and $(3b)$ instead of $(-3b)$.
• Option C: ${ }_{10} C_6(2 a)^6(-3 b)^4$ - This is incorrect because it uses $k=6$ and has incorrect powers.
• Option D: ${ }_{10} C_6(2 a)^4(-3 b)^6$ - This is incorrect because it uses $k=6$ and has incorrect powers.
• Option E: ${ }_{10} C_5(2 a)^5(-3 b)^5$ - This option correctly uses $k=5$, has $(2a)^5$ and $(-3b)^5$, which matches our derived expression.

So, the correct answer is E. ${ }_{10} C_5(2 a)^5(-3 b)^5$.

Common Mistakes to Avoid

When dealing with binomial expansions, it's easy to make a few common mistakes. Here are some tips to avoid them:

1. Incorrectly Identifying the Term Number: Remember that the first term corresponds to $k = 0$, not $k = 1$. So, the nth term corresponds to $k = n - 1$.
2. Forgetting the Negative Sign: If there's a negative sign in the binomial (like in our case with $-3b$), make sure to include it when raising to a power.
3. Mixing Up the Powers: Double-check that the powers of $x$ and $y$ add up to $n$. In the formula ${n \choose k} x^{n-k} y^k$, the sum of the exponents $(n-k) + k$ should always equal $n$.
4. Miscalculating the Binomial Coefficient: Be careful when calculating ${n \choose k}$. Use the formula $\frac{n!}{k!(n-k)!}$ and make sure to simplify correctly.

Real-World Applications

You might be wondering, "Where do we actually use binomial expansions in real life?" Well, there are several applications across various fields:

• Probability: Binomial expansions are used extensively in probability theory, especially when dealing with binomial distributions. For example, if you want to calculate the probability of getting a certain number of heads when flipping a coin multiple times, you'd use the binomial theorem.
• Statistics: In statistics, binomial expansions help in understanding sampling distributions and hypothesis testing.
• Computer Science: Binomial expansions can be used in algorithm analysis and in certain types of data compression.
• Engineering: Engineers use binomial expansions in various calculations, such as approximating complex functions and analyzing systems with multiple components.
• Finance: Actuaries and financial analysts use binomial models to price options and other financial derivatives.

Practice Problems

To solidify your understanding, here are a couple of practice problems:

1. Find the fourth term in the expansion of $(x + 2y)^7$.
2. Find the middle term in the expansion of $(3a - b)^8$.

Try solving these problems on your own, and feel free to ask if you get stuck! The more you practice, the more comfortable you'll become with binomial expansions.

Conclusion

So, there you have it! We've successfully identified the sixth term in the binomial expansion of $(2a - 3b)^{10}$. Remember to pay close attention to the term number, signs, and powers, and you'll be able to tackle these problems with confidence. Keep practicing, and you'll become a binomial expansion pro in no time! You got this, guys! Understanding these concepts not only helps in exams but also provides a solid foundation for more advanced topics in mathematics and various other fields. Happy expanding! And remember, practice makes perfect!