Coefficient Of The Third Term In Binomial Expansion Of (a+b)^6

The coefficient of the third term in the binomial expansion of $(a+b)^6$ is a common question in algebra. To fully grasp this, we need to delve into the binomial theorem and its applications. This article aims to provide a comprehensive explanation of how to find this coefficient, making the process clear and understandable. We will begin by revisiting the binomial theorem, then move on to identifying the third term, and finally, calculate its coefficient. This journey will not only answer the specific question but also equip you with the knowledge to tackle similar problems with confidence.

Unveiling the Binomial Theorem

At the heart of finding the coefficient lies the binomial theorem. This theorem provides a method for expanding expressions of the form $(a+b)^n$, where $n$ is a non-negative integer. The expansion results in a sum of terms, each consisting of a binomial coefficient, powers of $a$, and powers of $b$. The general form of the binomial theorem is given by:

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

Here, the binomial coefficient $\binom{n}{k}$ (read as "n choose k") is a crucial component. It represents the number of ways to choose $k$ elements from a set of $n$ elements and is calculated using the formula:

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

where $n!$ (n factorial) is the product of all positive integers up to $n$. Understanding this formula is the key to unlocking the coefficients in the binomial expansion. Let's break down each component to ensure a solid foundation. The factorial notation, $n!$, signifies the product of all positive integers less than or equal to $n$. For instance, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. Mastering the calculation of factorials is essential for determining binomial coefficients. The binomial coefficient $\binom{n}{k}$ can also be visualized using Pascal's Triangle, where each number is the sum of the two numbers directly above it. This visual representation offers an alternative method for finding binomial coefficients, particularly for smaller values of $n$.

Identifying the Third Term in the Expansion

Now, let's apply the binomial theorem to our specific case, $(a+b)^6$. We need to pinpoint the third term in the expansion. In the binomial expansion, the terms are indexed starting from $k=0$. Thus, the first term corresponds to $k=0$, the second term to $k=1$, and the third term to $k=2$. This indexing is vital for correctly identifying the term we need to focus on. Using the general form of the binomial theorem, the third term (when $k=2$) can be expressed as:

$\binom{6}{2} a^{6-2} b^2 = \binom{6}{2} a^4 b^2$

This expression tells us that the third term will involve the binomial coefficient $\binom{6}{2}$, $a$ raised to the power of 4, and $b$ squared. The next step is to calculate the binomial coefficient to find the numerical value associated with this term. The structure of the binomial expansion reveals a pattern in the powers of $a$ and $b$. As we move from one term to the next, the power of $a$ decreases by one, while the power of $b$ increases by one. This pattern is a direct consequence of the binomial theorem's formula and provides a useful check for our calculations. Recognizing this pattern can help in predicting and verifying the terms in the expansion.

Calculating the Coefficient

The final step is to calculate the binomial coefficient $\binom{6}{2}$. Using the formula:

$\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(4 \times 3 \times 2 \times 1)}$

We can simplify this expression:

$\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15$

Thus, the coefficient of the third term in the binomial expansion of $(a+b)^6$ is 15. This calculation showcases the power of the binomial coefficient in determining the numerical factor of each term in the expansion. Let's break down the calculation step by step to ensure clarity. First, we expand the factorials: $6! = 720$, $2! = 2$, and $4! = 24$. Then, we substitute these values into the formula: $\binom{6}{2} = \frac{720}{2 \times 24}$. Simplifying the denominator gives us $\binom{6}{2} = \frac{720}{48}$. Finally, dividing 720 by 48 yields the result 15. This methodical approach minimizes the chances of errors and ensures an accurate calculation of the binomial coefficient.

Conclusion: The Coefficient Unveiled

In conclusion, the coefficient of the third term in the binomial expansion of $(a+b)^6$ is 15. This result was obtained by applying the binomial theorem, identifying the relevant term, and calculating the binomial coefficient. The binomial theorem is a powerful tool in algebra, enabling us to expand expressions and find specific terms with ease. Understanding the underlying principles and formulas is crucial for mastering this concept. The ability to calculate binomial coefficients and apply the binomial theorem is a valuable skill in various mathematical contexts. This skill extends beyond simple expansions and finds applications in probability, statistics, and other advanced topics. By understanding the binomial theorem, you gain a deeper appreciation for the patterns and relationships within algebraic expressions. The process of identifying and calculating coefficients becomes more intuitive, paving the way for tackling more complex problems with confidence. Practice is key to mastering the binomial theorem. By working through various examples and exercises, you can solidify your understanding and develop fluency in applying the theorem. This mastery will not only benefit your algebra skills but also provide a solid foundation for future mathematical endeavors.