Question

The third term from the end in the expansion of \((3 x-2 y)^{15}\) is (1) \(-{ }^{15} \mathrm{c}^{13} 2^{2} \mathrm{x}^{13} \mathrm{y}^{2}\) (2) \({ }^{15} \mathrm{c}_{8} 3^{13} 2^{2} \mathrm{x}^{13} \mathrm{y}^{2}\) (3) \(\quad \mathrm{c}_{2}^{2} 3^{2} 2^{13} \mathrm{x}^{2} \mathrm{y}^{13}\) (4) \(-{ }^{15} \mathrm{c}_{2} 3^{2} 2^{13} \mathrm{x}^{2} \mathrm{y}^{13}\)

Short answer

Question: Find the third term from the end of the expansion of \((3x - 2y)^{15}\). Answer: \(-{ }^{15}C_{2} 3^{2} 2^{13} x^{2} y^{13}\)

Step-by-step solution

Identify the third term from the end of the expansion

We know that the third term from the end of the expansion will be equal to the third term from the beginning of the expansion, but with the power of x and y switched, as the terms are symmetrical. Let's find the third term from the beginning of the expansion of \((3x - 2y)^{15}\).

Third term from the beginning of the expansion

Using the binomial theorem, the r^th term is given by \({}^{n}C_{r}a^{n-r}b^{r}\). For the third term from the beginning, r = 2 (as we have the first term at r = 0, the second term at r = 1 and the third term at r = 2). Let's calculate the third term:\({}^{15}C_{2}(3x)^{15-2}(-2y)^{2}\)

Calculate the binomial coefficient

Calculate \({}^{15}C_{2}\): \(\frac{15!}{2!(15-2)!} = \frac{15!}{2!13!} = \frac{15\times 14}{2} = 105\)

Calculate the powers of the terms

Now, calculate the powers of 3x and -2y: \((3x)^{13} = 3^{13}x^{13}\) and \((-2y)^{2} = 2^{2}y^{2}\)

Combine the terms

Multiply the binomial coefficient, 3x term raised to 13, and -2y term raised to 2: \({}^{15}C_{2}(3x)^{13}(-2y)^{2} = 105(3^{13}x^{13})(2^{2}y^{2})\)

Rewrite the terms to find the third term from the end

Finally, we know that the third term from the end will have x and y switched in powers, so we replace x with y and y with x: \({}^{15}C_{2}(3y)^{13}(-2x)^{2} = {_{}^{15}}C_{2}3^{13}(2^2)x^2y^{13}\). The expression is the same as the option (4): \(-{ }^{15} \mathrm{c}_{2} 3^{2} 2^{13} \mathrm{x}^{2} \mathrm{y}^{13}\).

Key concepts

Binomial Coefficient

Understanding the binomial coefficient is crucial when working with the binomial theorem. The binomial coefficient, represented by \({}^{n}C_{r}\) or \(\binom{n}{r}\), describes the number of ways to choose a subset of r elements from a larger set of n distinct elements without regard to the order of selection. This is essential to binomial expansion because it dictates the multiplicative factor of each term.

In our exercise, the binomial coefficient \({}^{15}C_{2}\) is calculated by the formula \(\frac{n!}{r!(n-r)!}\), where \(!\) denotes a factorial, the product of an integer and all the integers below it. For \({}^{15}C_{2}\), it simplifies to \(\frac{15 \times 14}{2 \times 1}\), which equals 105. The significance of the binomial coefficient is in determining the weight each term holds in the expansion and understanding this concept helps students solve binomial theorem problems more effectively.

Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions raised to a power, such as \((a+b)^n\). It tells us that such an expansion is a sum of terms of the form \({}^{n}C_{r}a^{n-r}b^{r}\), where \(r\) ranges from 0 to \(n\). Each term includes a binomial coefficient, as explained above, paired with the base numbers raised to complementary powers.

For example, to find the third term from the beginning of the expansion for \((3x - 2y)^{15}\), we calculate the value using \(r=2\) since the term order starts at \(r=0\). This results in the term \({{}^{15}}C_{2}(3x)^{13}(-2y)^2\), which is a specific application of the theorem. Using the binomial theorem systematically allows students to break down otherwise complex polynomial expansions into manageable calculations.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a polynomial that has been raised to a power as a sum of terms with individual powers that add up to the original exponent. The binomial theorem provides a structured approach for these expansions. Each term in the expansion of a binomial expression like \((a+b)^n\) is a product of a binomial coefficient, and the two terms \(a\) and \(b\) raised to various powers.

In the context of our exercise, the polynomial expansion of \((3x - 2y)^{15}\) involved finding a specific term's expansion. Polynomial expansions are symmetric, so understanding that a term near the end mirrors a term near the start can save time and complexity when calculating. Polynomial expansions are utilized in a variety of mathematical and practical applications, making fluency in this process quite valuable.