Chapter 8.4, Problem 70E

To determine

To find: the coefficient of $x^4{y}^4$ in the expansion of given binomial.

Answer to Problem 70E

The coefficient of $x^4{y}^4$ in the expansion of given binomial is $90720$ .

Explanation of Solution

Given information:

Given binomial

  ${\left(2x-3y\right)}^8$

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.

  ${\left(2x-3y\right)}^8$

Let ${\left(r+1\right)}^{th}$ term isof $x^4{y}^4$ .

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

  $n=8,y=-3y,x=2x$

Now, calculate the ${\left(r+1\right)}^{th}$ term of the expansion as shown:

  $\begin{array}{c}C{}_r{x}^{n-r}{y}^r=C{}_r{\left(2x\right)}^{8-r}{\left(-3y\right)}^r\\ =C{}_r\left(2^{8-r}\right)x^{8-r}{\left(-3\right)}^r{y}^r\end{array}$

Again, ${\left(x\right)}^{8-r}{y}^r=x^4{y}^4$

Therefore,

  $\begin{array}{c}8-r=4\\ r=4\end{array}$

Now, the 5thterm is the term of $x^4{y}^4$ .

Coefficient of 5^thterm can be calculated as shown:

  $\begin{array}{c}C{}_4{\left(2\right)}^{8-4}{\left(-3\right)}^4=\frac{8!}{4!4!}\times 2^4\times 3^4\\ =\frac{8\times 7\times 6\times 5\times 4!}{=4!4\times 3\times 2\times 1}\times 16\times 81\\ =70\times 16\times 81\\ =90720\end{array}$

Hence, the coefficient of $x^4{y}^4$ in the expansion of given binomial is $90720$ .