Chapter 8.4, Problem 63E

To determine

To find: the 10^thterm in the expansion of given binomial.

Answer to Problem 63E

The 10^thterm in the expansion of given binomial is $-4.33026\times {10}^9{x}^3{y}^9$ .

Explanation of Solution

Given information:

Given binomial

  ${\left(10x-3y\right)}^{12}$

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.

  ${\left(10x-3y\right)}^{12}$

Now, for 10^thterm compare the above binomial with ${\left(x+y\right)}^n$ .

  $n=12,r+1=9,y=-3y,x=10x$

So, $r=8$

Now, calculate the 10^th term of the expansion as shown:

  $\begin{array}{c}C{}_r{x}^{n-r}{y}^r=C{}_9{\left(10x\right)}^{12-9}{\left(-3y\right)}^9\\ =\frac{12!}{\left(12-9\right)!9!}1000x^3\left(-19683y^9\right)\\ =\frac{12!}{3!9!}1000x^3\left(-19683y^9\right)\\ =\frac{12\times 11\times 10\times 9!}{9!3\times 2\times 1}1000x^3\left(-19683y^9\right)\\ =-4.33026\times {10}^9{x}^3{y}^9\end{array}$

Hence, the 10^thterm in the expansion of given binomial is $-4.33026\times {10}^9{x}^3{y}^9$ .