Chapter 9.5, Problem 49E

To determine

The co-efficient $a$ of the term $a{x}^2{y}^8$ in the Binomial expansion of ${\left(4x-y\right)}^{10}$ .

Answer to Problem 49E

The co-efficient $a$ of the term $a{x}^2{y}^8$ is $720$ .

Explanation of Solution

Given information :

Given expression is ${\left(4x-y\right)}^{10}$

We know that the ${\left(r+1\right)}^{th}$ term of the Binomial expansion of ${\left(x+y\right)}^n$ is ${}_n{C}_r{x}^{n-r}{y}^r$ ,

Thus, the ${\left(r+1\right)}^{th}$ term of the Binomial expansion of ${\left(4x-y\right)}^{10}$ is $10C_r{\left(4x\right)}^{10-r}{\left(-y\right)}^r\left(\because n=10,x=4x,y=-y\right)$

Now replace $r=8$ , to get $x^2{y}^8$ in above expression

  $\begin{array}{l}{\Rightarrow }_{10}{C}_8{\left(4x\right)}^{10-8}{\left(-y\right)}^8\\ \Rightarrow \frac{10!}{\left(2\right)!\left(8!\right)}\times 4^2{x}^2{\left(y\right)}^8\\ \Rightarrow \frac{10\times 9\times 8!}{\left(2\times 1\right)\times \left(8!\right)}\times 16\times x^2{\left(y\right)}^8\\ \Rightarrow 90\times 8x^2{y}^8\\ \therefore 720x^2{y}^8\end{array}$

Hence, the co-efficient $a$ of the term $a{x}^2{y}^8$ is $720$ .