Chapter 8.4, Problem 69E

To determine

To find: the coefficient of $x^8{y}^6$ in the expansion of given binomial.

Answer to Problem 69E

The coefficient of $x^8{y}^6$ in the expansion of given binomial is $210$ .

Explanation of Solution

Given information:

Given binomial

  ${\left(x^2+y\right)}^{10}$

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.

  ${\left(x^2+y\right)}^{10}$

Let ${\left(r+1\right)}^{th}$ term is of $x^8{y}^6$ .

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

  $n=10,x=x^2$

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

  $\begin{array}{c}C{}_r{x}^{n-r}{y}^r=C{}_r{\left(x^2\right)}^{10-r}{\left(y\right)}^r\\ =C{}_r{\left(x\right)}^{20-2r}{y}^r\end{array}$

Again, ${\left(x\right)}^{20-2r}{y}^r=x^8{y}^6$

Therefore,

  $\begin{array}{c}20-2r=8\\ r=6\end{array}$

Now, the 7^thterm is the term of $x^8{y}^6$ .

Coefficient of 7^thterm can be calculated as shown:

  $\begin{array}{c}C{}_6=\frac{10!}{6!4!}\times 4\\ =\frac{10\times 9\times 8\times 7\times 6!}{6!4\times 3\times 2\times 1}\\ =210\end{array}$

Hence, the coefficient of $x^8{y}^6$ in the expansion of given binomial is $210$ .