Finding The Term 45x^8y^2 In (x+y)^10 Using The Binomial Theorem

The binomial theorem stands as a cornerstone of algebraic expansion, providing a systematic method for expressing powers of binomials—expressions with two terms—as sums of individual terms. This theorem is not just a mathematical curiosity; it has profound implications across various fields, including probability, statistics, and even computer science. Understanding the binomial theorem unlocks the ability to predict and analyze patterns within binomial expansions, making it an invaluable tool for problem-solving and mathematical exploration. In this article, we embark on a journey to unravel the specific term containing $45x^8y^2$ within the expansion of $(x+y)^{10}$, showcasing the power and elegance of the binomial theorem.

The binomial theorem allows us to expand expressions of the form $(a + b)^n$, where 'n' is a non-negative integer. The expansion results in a sum of terms, each with a specific coefficient and powers of 'a' and 'b'. The beauty of the theorem lies in its ability to predict these coefficients and powers without requiring tedious multiplication. The general formula for the binomial theorem is:

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

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

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

The exclamation mark denotes the factorial, where $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$. The binomial coefficient tells us the number of ways to choose 'k' items from a set of 'n' items, and it plays a crucial role in determining the coefficients in the binomial expansion.

Each term in the expansion corresponds to a specific value of 'k', ranging from 0 to 'n'. The term with $k = 0$ gives us the first term, the term with $k = 1$ gives us the second term, and so on. The powers of 'a' and 'b' in each term are determined by 'n' and 'k', with the power of 'a' decreasing from 'n' to 0 and the power of 'b' increasing from 0 to 'n'.

To apply the binomial theorem effectively, it's essential to understand the notation and the components of the formula. The binomial coefficient, in particular, is a key element in determining the magnitude of each term in the expansion. By understanding how factorials and combinations work, we can efficiently calculate these coefficients and predict the structure of the expanded form.

To effectively apply the binomial theorem, let's break down the process into a series of steps. This step-by-step guide will illuminate how to identify specific terms within a binomial expansion and calculate their coefficients.

1. Understanding the General Term:

The heart of the binomial theorem lies in the general term formula. This formula allows us to directly calculate any specific term in the expansion without having to compute all the preceding terms. The general term, denoted as $T_{k+1}$, is given by:

$T_{k+1} = \binom{n}{k} a^{n-k} b^k$

Here, 'n' represents the power to which the binomial is raised, 'k' is the term number minus 1 (starting from 0), 'a' and 'b' are the terms within the binomial, and $\binom{n}{k}$ is the binomial coefficient.

2. Identifying 'n', 'a', and 'b':

The first step in applying the binomial theorem is to correctly identify the values of 'n', 'a', and 'b' from the given binomial expression. In our case, we have $(x+y)^{10}$. Therefore, $n = 10$, $a = x$, and $b = y$. This identification is crucial as these values will be directly substituted into the general term formula.

3. Determining the Value of 'k':

To find the term with $45x^8y^2$, we need to determine the value of 'k' that corresponds to this term. Observe that the power of 'x' is 8 and the power of 'y' is 2. From the general term formula, we know that the power of 'x' is $n-k$ and the power of 'y' is 'k'. Thus, we have:

$n - k = 8$

Since $n = 10$, we can solve for 'k':

$10 - k = 8$

$k = 2$

This value of $k = 2$ tells us that the term we are looking for is the third term in the expansion (since the general term is indexed as $T_{k+1}$).

4. Calculating the Binomial Coefficient:

Now that we have 'n' and 'k', we can calculate the binomial coefficient $\binom{n}{k}$. In our case, we need to calculate $\binom{10}{2}$:

$\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1} = 45$

This binomial coefficient of 45 is a key component of the term we are seeking.

5. Constructing the Term:

With all the pieces in place, we can now construct the term. Substitute $n = 10$, $k = 2$, $a = x$, and $b = y$ into the general term formula:

$T_{2+1} = T_3 = \binom{10}{2} x^{10-2} y^2 = 45x^8y^2$

Thus, the term containing $45x^8y^2$ is indeed the third term in the expansion of $(x+y)^{10}$.

Having dissected the binomial theorem and applied it step-by-step, we arrive at the solution to our initial question: identifying the term with $45x^8y^2$ in the expansion of $(x+y)^{10}$.

Through our calculations, we determined that the value of 'k' corresponding to this term is 2. This means that the term we are looking for is the third term in the expansion, denoted as $T_3$. We calculated the binomial coefficient $\binom{10}{2}$ to be 45, which matches the coefficient in the term $45x^8y^2$.

Therefore, the $\boxed{3rd}$ term of the expanded form of the given binomial expression is $45x^8y^2$. This result highlights the power of the binomial theorem in precisely identifying specific terms within a complex expansion. The theorem not only provides a formula for calculating these terms but also offers a systematic approach to understanding the structure and patterns within binomial expansions.

The binomial theorem is a powerful tool for expanding binomial expressions and identifying specific terms within the expansion. By understanding the general term formula and following a step-by-step approach, we can efficiently determine the term with a given coefficient and powers of the variables. In this case, we successfully identified the third term as the term containing $45x^8y^2$ in the expansion of $(x+y)^{10}$. This exercise demonstrates the practical application of the binomial theorem and its importance in algebraic manipulations.

The binomial theorem extends far beyond simple algebraic expansions; it serves as a foundational concept with diverse applications across various branches of mathematics and other scientific disciplines. Understanding its implications enriches our mathematical toolkit and provides a deeper appreciation for its versatility.

1. Probability Theory and Statistics:

The binomial theorem plays a crucial role in probability theory, particularly in the context of binomial distributions. A binomial distribution describes the probability of obtaining a certain number of successes in a sequence of independent trials, each with a fixed probability of success. The binomial coefficients, derived from the binomial theorem, directly correspond to the probabilities in a binomial distribution. For example, when calculating the probability of getting exactly 'k' heads in 'n' coin flips, the binomial coefficient $\binom{n}{k}$ is a key component of the formula. This connection highlights the theorem's practical relevance in statistical analysis and probability calculations.

2. Combinatorics and Counting Problems:

The binomial coefficients themselves have deep roots in combinatorics, the branch of mathematics concerned with counting and arrangements. The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose 'k' objects from a set of 'n' distinct objects, without regard to order. This concept is fundamental to solving a wide range of counting problems, such as determining the number of possible committees that can be formed from a group of people or calculating the number of ways to arrange letters in a word. The binomial theorem provides a powerful framework for understanding and solving these types of combinatorial problems.

3. Calculus and Series Expansions:

The binomial theorem can be extended to cases where the exponent is not a positive integer, leading to the binomial series. This series provides a way to represent functions as infinite sums, which is a cornerstone of calculus. For instance, the binomial series can be used to approximate values of functions like $(1+x)^r$ for any real number 'r', even when 'x' is not small. This extension of the theorem opens doors to advanced mathematical techniques and applications in areas like numerical analysis and approximation theory.

4. Computer Science and Algorithm Analysis:

In computer science, the binomial theorem and binomial coefficients arise in various contexts, including algorithm analysis and data structures. For example, the number of possible subsets of a set with 'n' elements is $2^n$, which can be derived from the binomial theorem by summing the binomial coefficients $\binom{n}{k}$ for all values of 'k' from 0 to 'n'. This relationship is useful in analyzing the complexity of algorithms and understanding the behavior of data structures. The theorem's connection to combinatorial problems also makes it relevant in areas like cryptography and coding theory.

5. Physics and Engineering:

In physics and engineering, the binomial theorem finds applications in approximating physical quantities and modeling various phenomena. For example, in optics, the theorem can be used to approximate the behavior of lenses and mirrors. In fluid mechanics, it can be used to simplify complex equations involving pressure and velocity. The theorem's ability to provide accurate approximations for expressions involving powers makes it a valuable tool for engineers and physicists.

In conclusion, our exploration of the binomial expression $(x+y)^{10}$ and the identification of the term $45x^8y^2$ has underscored the power and versatility of the binomial theorem. This theorem is not merely a formula for expanding binomials; it is a fundamental concept that permeates various branches of mathematics and science. From its roots in combinatorics and algebra to its applications in probability, statistics, calculus, computer science, and physics, the binomial theorem stands as a testament to the interconnectedness of mathematical ideas.

By understanding the binomial theorem, we gain access to a powerful tool for solving a wide range of problems, from calculating probabilities to approximating complex functions. Its elegant formula and systematic approach provide a framework for understanding patterns within binomial expansions and making predictions about the behavior of systems governed by these patterns. As we have seen, the binomial theorem is not just a theoretical construct; it is a practical tool with real-world applications that continue to shape our understanding of the world around us. The enduring legacy of the binomial theorem lies in its ability to connect seemingly disparate mathematical concepts and provide a unifying framework for problem-solving and mathematical exploration.