Chapter 8.4, Problem 65E

To determine

To find: the coefficient of $x^5$ in the expansion of given binomial.

Answer to Problem 65E

The coefficient of $x^5$ in the expansion of given binomial is $1732104$ .

Explanation of Solution

Given information:

Given binomial

  ${\left(x+3\right)}^{12}$

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.

  ${\left(x+3\right)}^{12}$

Let ${\left(r+1\right)}^{th}$ term has the term $x^5$ .

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

  $n=12,y=3$

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

  $C{}_r{x}^{n-r}{y}^r=C{}_r{\left(x\right)}^{12-r}{\left(3\right)}^r$

Again, ${\left(x\right)}^{12-r}=x^5$

Therefore,

  $\begin{array}{c}12-r=5\\ r=7\end{array}$

Now, the 8^th term is the term of $x^5$ .

Coefficient of 8^th term can be calculated as shown:

  $\begin{array}{c}C{}_7{\left(3\right)}^7=\frac{12!}{5!7!}\times 3^7\\ =\frac{12\times 11\times 10\times 9\times 8\times 7!}{7!5\times 4\times 3\times 2\times 1}\times 3^7\\ =11\times 9\times 8\times 3^7\\ =1732104\end{array}$

Hence, the coefficient of $x^5$ in the expansion of given binomial is $1732104$ .