Chapter 8.4, Problem 59E

To determine

To find: the 3rd term in the expansion of given binomial.

Answer to Problem 59E

The 3^rdterm in the expansion of given binomial is $360x^3{y}^2$ .

Explanation of Solution

Given information:

Given binomial

  ${\left(x-6y\right)}^5$

Formula used:

  ${\left(r+1\right)}^{th}$ term in the expansion of a binomial ${\left(x+y\right)}^n$ is given by $C{}_r{x}^{n-r}{y}^r$

And the coefficient of

  $x^{n-r}{y}^r$ is ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Calculation:

Consider the given binomial.

  ${\left(x-6y\right)}^5$

Now, for 3^rd term compare the above binomial with ${\left(x+y\right)}^n$ .

  $n=5,r+1=3,y=-6y$

So, $r=2$

Now, calculate the 3^rdterm of the expansion as shown:

  $\begin{array}{c}C{}_r{x}^{n-r}{y}^r=C{}_2{x}^{5-2}{\left(-6y\right)}^2\\ =\frac{5!}{\left(5-2\right)!2!}{x}^{3}36y^2\\ =\frac{5!}{2!3!}{x}^{3}36y^2\\ =\frac{5\times 4\times 3!}{2!3!}{x}^{3}36y^2\\ =360x^3{y}^2\end{array}$

Hence, the 3^rdterm in the expansion of given binomial is $360x^3{y}^2$ .