Finding The Coefficient B In Binomial Expansion Of (x+y)^10

Hey guys! Let's dive into a cool math problem today. We're going to figure out how to find a specific coefficient in a binomial expansion. Binomial expansions might sound intimidating, but they're actually pretty straightforward once you get the hang of them. We'll take it step by step, so no worries if it seems confusing at first. Our main goal here is to not only solve the problem but also to understand the underlying concepts. This way, you’ll be ready to tackle similar problems with confidence. So, let’s get started and make math a little less mysterious!

Understanding Binomial Expansion

Okay, so what exactly is a binomial expansion? Simply put, it’s what happens when you raise a binomial (an expression with two terms, like x + y) to a power. For example, $(x + y)^2$ is a binomial squared. Expanding it means multiplying it out: $(x + y) * (x + y)$, which gives you $x^2 + 2xy + y^2$. But what happens when you have a higher power, like $(x + y)^{10}$? You could multiply it out by hand, but that would take ages! That's where the binomial theorem comes in handy. The binomial theorem gives us a formula to find the coefficients in these expansions quickly. The coefficients are the numbers that multiply the terms (like the '2' in $2xy$).

The Binomial Theorem: Your New Best Friend

The binomial theorem states that for any non-negative integer n, the expansion of $(x + y)^n$ can be written as:

$(x + y)^n = \binom{n}{0}x^n y^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + ... + \binom{n}{n}x^0 y^n$

Whoa, that looks complicated, right? Let’s break it down. The funny-looking symbols like $\binom{n}{k}$ are called binomial coefficients, and they're read as "n choose k". They tell you the coefficient of each term in the expansion. The formula to calculate them is:

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

Where "!" means factorial. For example, 5! (5 factorial) is 5 * 4 * 3 * 2 * 1 = 120. So, to find a specific coefficient, you just plug in the values of n (the power) and k (the term number, starting from 0).

Pascal's Triangle: A Visual Aid

Now, calculating those factorials can be a bit tedious, especially for larger numbers. That’s where Pascal’s Triangle comes to the rescue! 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 binomial expansions. The top row (row 0) corresponds to $(x + y)^0$, the second row (row 1) corresponds to $(x + y)^1$, and so on. Each number in a row gives you the coefficient for that term. For example, the row for $(x + y)^2$ is 1 2 1, which matches the coefficients in the expansion $x^2 + 2xy + y^2$.

Solving the Problem: Finding the Value of B

Alright, let’s get back to our original problem. We're given the coefficients for the expansion of $(x + y)^{10}$: 1, 10, 45, 120, 210, 252, 210, B, 45, 10, 1. We need to find the value of B. Since this is the expansion of something to the power of 10, we know that n = 10.

Using Symmetry to Our Advantage

One cool thing about binomial coefficients is that they're symmetrical. This means that $\binom{n}{k}$ is the same as $\binom{n}{n-k}$. In the context of our problem, this means the coefficients in the expansion are symmetrical. Notice how the sequence starts with 1, 10, 45, and so on, and then mirrors itself at the end with 45, 10, 1. The middle term is 252, and the term we're looking for, B, is right after that.

Since the coefficients are symmetrical, we can count from either end to find the position of B. The coefficients given are for the terms corresponding to k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. B is in the eighth position (corresponding to k = 7). So, B is actually $\binom{10}{7}$.

Calculating the Binomial Coefficient

Now, let's calculate $\binom{10}{7}$ using the formula:

$\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!}$

Let's break down those factorials:

• 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
• 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
• 3! = 3 * 2 * 1 = 6

We can simplify the expression by canceling out the 7! in the numerator and denominator:

$\binom{10}{7} = \frac{10 * 9 * 8}{3 * 2 * 1} = \frac{10 * 9 * 8}{6}$

Now, simplify further:

$\binom{10}{7} = 10 * 3 * 4 = 120$

So, the value of B is 120.

Why is B = 120?

Alright, let’s recap why we found B to be 120. We started by understanding what a binomial expansion is and how the binomial theorem helps us find the coefficients. We learned about Pascal’s Triangle as a visual tool to identify these coefficients. Then, we applied the binomial coefficient formula to directly calculate the value. The key here was recognizing the symmetry in binomial coefficients, which allowed us to identify B’s position in the expansion.

We used the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ and plugged in the values n = 10 and k = 7, corresponding to the eighth term in the expansion. After calculating the factorials and simplifying, we arrived at the result: $\binom{10}{7} = 120$. Thus, B is indeed 120. This step-by-step approach ensures we understand not only the answer but also the process behind it. Remember, guys, math isn't just about getting the right answer; it’s about understanding why that answer is correct.

Alternative Method: Using Pascal's Triangle

If we didn't want to calculate factorials, we could also use Pascal's Triangle. We'd need to look at the 11th row (remember, we start counting rows from 0, so row 10 corresponds to the expansion of $(x + y)^{10}$). The 11th row of Pascal's Triangle is: 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. You can see that the eighth number in this row (corresponding to k = 7) is indeed 120. So, Pascal’s Triangle provides a quick visual confirmation of our result. Pretty neat, huh?

Key Takeaways

So, what did we learn today? A whole bunch of cool stuff about binomial expansions!

• Binomial Theorem: This is your go-to tool for expanding binomials raised to a power. Remember the formula: $(x + y)^n = \binom{n}{0}x^n y^0 + \binom{n}{1}x^{n-1}y^1 + ... + \binom{n}{n}x^0 y^n$.
• Binomial Coefficients: These tell you the coefficients of each term in the expansion. The formula to calculate them is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
• Pascal's Triangle: This is a handy visual aid to find binomial coefficients. Each number is the sum of the two numbers directly above it.
• Symmetry: Binomial coefficients are symmetrical, meaning $\binom{n}{k}$ is the same as $\binom{n}{n-k}$. This can save you time when solving problems.

Practice Makes Perfect

The best way to get comfortable with binomial expansions is to practice. Try expanding different binomials with different powers. Use both the formula and Pascal's Triangle to check your work. You'll be a pro in no time! Also, remember to always double-check your work and ensure each step makes sense. Math is like building blocks; each concept builds on the previous one.

Final Thoughts

And there you have it! We've successfully found the value of B in the binomial expansion. We tackled the problem using the binomial theorem, calculated factorials, and even peeked at Pascal's Triangle. I hope this explanation has made binomial expansions a little less scary and a lot more interesting. Remember, guys, math can be fun if you approach it with the right mindset and a willingness to learn. Keep practicing, keep exploring, and who knows? Maybe you'll discover the next big mathematical breakthrough! Keep up the great work, and I’ll catch you in the next math adventure!