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

Aug 3, 2025 by Omar Yusuf 60 views

Unveiling the Coefficient of the Third Term in the Binomial Expansion of (a+b)^6

Hey guys! Let's dive into the fascinating world of binomial expansions and tackle a common question that often pops up in mathematics: What is the coefficient of the third term in the binomial expansion of (a+b)^6? This might sound a bit intimidating at first, but trust me, it's a lot easier than it looks. We'll break it down step by step, making sure you not only get the answer but also understand the underlying principles. So, grab your thinking caps, and let's get started!

Understanding Binomial Expansion

Before we jump into the specifics, let's quickly recap what binomial expansion is all about. In essence, binomial expansion is a method of expanding expressions of the form (a + b)^n, where 'n' is a non-negative integer. When we expand such an expression, we get a series of terms, each with its own coefficient and powers of 'a' and 'b'.

Think of it like this: when you multiply (a + b) by itself multiple times, you're essentially combining different combinations of 'a' and 'b'. The binomial theorem provides us with a systematic way to figure out what those combinations are and what their corresponding coefficients should be.

The general formula for the binomial theorem is given by:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

where 'k' ranges from 0 to n, and "(n choose k)" represents the binomial coefficient, which is calculated as:

(n choose k) = n! / (k! * (n-k)!)

Here, '!' denotes the factorial operation (e.g., 5! = 5 * 4 * 3 * 2 * 1). The binomial coefficient tells us how many ways we can choose 'k' items from a set of 'n' items, without regard to order. This is crucial in determining the coefficients in our binomial expansion.

Key Concepts to Remember

Binomial Theorem: The foundation for expanding expressions of the form (a + b)^n.
Binomial Coefficient: Denoted as (n choose k), it represents the number of ways to choose 'k' items from 'n' items.
Factorial: Denoted by '!', it represents the product of all positive integers up to a given number.

Identifying the Third Term

Now that we have a solid understanding of binomial expansion, let's focus on identifying the third term in the expansion of (a + b)^6. Remember, the terms in the expansion are numbered starting from 0. So, the first term corresponds to k = 0, the second term corresponds to k = 1, and the third term corresponds to k = 2.

Using the binomial theorem formula, the third term can be written as:

(6 choose 2) * a^(6-2) * b^2

This is where the binomial coefficient comes into play. We need to calculate (6 choose 2) to find the coefficient of the third term. Let's break it down:

(6 choose 2) = 6! / (2! * (6-2)!)

(6 choose 2) = 6! / (2! * 4!)

(6 choose 2) = (6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (4 * 3 * 2 * 1))

We can simplify this by canceling out the common factors:

(6 choose 2) = (6 * 5) / (2 * 1)

(6 choose 2) = 30 / 2

(6 choose 2) = 15

So, the binomial coefficient (6 choose 2) is equal to 15. This means that the coefficient of the third term in the expansion of (a + b)^6 is 15. Now we have all the pieces we need to construct the third term:

Third term = 15 * a^(6-2) * b^2

Third term = 15 * a^4 * b^2

Therefore, the third term in the binomial expansion of (a + b)^6 is 15a^4b2, and its coefficient is 15.

Calculating the Coefficient

To solidify our understanding, let's walk through the calculation of the coefficient again, emphasizing the key steps. As we established earlier, the coefficient of the third term in the binomial expansion of (a + b)^6 is given by the binomial coefficient (6 choose 2).

The formula for the binomial coefficient is:

(n choose k) = n! / (k! * (n-k)!)

In our case, n = 6 (the exponent in (a + b)^6) and k = 2 (since we're looking for the third term, which corresponds to k = 2). Plugging these values into the formula, we get:

(6 choose 2) = 6! / (2! * (6-2)!)

Now, let's expand the factorials:

(6 choose 2) = (6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (4 * 3 * 2 * 1))

Notice that we have 4! (4 * 3 * 2 * 1) in both the numerator and the denominator. We can cancel these out to simplify the expression:

(6 choose 2) = (6 * 5) / (2 * 1)

Now, it's just a matter of simple arithmetic:

(6 choose 2) = 30 / 2

(6 choose 2) = 15

And there you have it! The coefficient of the third term in the binomial expansion of (a + b)^6 is 15. This result confirms our earlier calculation and reinforces the importance of understanding the binomial coefficient in binomial expansion problems.

Alternative Approach: Pascal's Triangle

For those who prefer a more visual approach, Pascal's Triangle provides an alternative method for finding binomial coefficients. 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 values of 'n' in the binomial expansion (starting from n = 0), and the numbers in each row represent the binomial coefficients (n choose k) for k = 0, 1, 2, ..., n.

To find the coefficient of the third term in the expansion of (a + b)^6, we need to look at the row corresponding to n = 6 in Pascal's Triangle. The first few rows of Pascal's Triangle look like this:

        1
      1   1
    1   2   1
  1   3   3   1
1   4   6   4   1

To get the row for n = 6, we continue the pattern:

1   5   10  10  5   1

And finally:

1   6   15  20  15  6   1

The row corresponding to n = 6 is 1 6 15 20 15 6 1. These numbers represent the binomial coefficients (6 choose 0), (6 choose 1), (6 choose 2), (6 choose 3), (6 choose 4), (6 choose 5), and (6 choose 6), respectively. Since we're looking for the coefficient of the third term, which corresponds to k = 2, we look at the third number in the row (remembering that we start counting from 0), which is 15. This confirms our earlier calculation using the binomial coefficient formula.

Using Pascal's Triangle can be a quicker and more intuitive way to find binomial coefficients, especially for smaller values of 'n'. However, for larger values of 'n', the binomial coefficient formula becomes more practical.

The Answer and Its Significance

So, after our journey through binomial expansion and coefficient calculations, we've arrived at the answer: the coefficient of the third term in the binomial expansion of (a + b)^6 is 15. This corresponds to answer choice B. But more importantly than just getting the right answer, we've gained a deeper understanding of the binomial theorem and how to apply it.

The binomial theorem is a powerful tool in mathematics, with applications ranging from probability and statistics to calculus and algebra. Understanding how to expand binomials and calculate coefficients is a fundamental skill that will serve you well in your mathematical endeavors. The coefficient itself tells us the numerical factor that multiplies the corresponding powers of 'a' and 'b' in that term. In this specific case, the third term is 15a^4b2, indicating that this particular combination of a^4 and b^2 appears 15 times in the expansion.

Wrapping Up

I hope this explanation has helped you grasp the concept of binomial expansion and how to find specific coefficients within it. Remember, the key is to understand the binomial theorem, the binomial coefficient, and how they relate to each other. Whether you prefer the formulaic approach or the visual aid of Pascal's Triangle, you now have the tools to tackle similar problems with confidence.

Keep practicing, keep exploring, and most importantly, keep enjoying the beauty of mathematics! You've got this! Now you understand clearly what is the coefficient of the third term in the binomial expansion of (a+b)^6?