Chapter 12, Problem 37SGA

To determine

To find: the 5^th term of the binomial.

Answer to Problem 37SGA

  $3360x^6$

Explanation of Solution

Given information:

  ${\left(x-2\right)}^{10}$

Calculation:

For positive integer $n$ , then the following will hold true.

  ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\frac{n\left(n-1\right)x^{n-2}{y}^2}{\left(1\right)\left(2\right)}+\frac{n\left(n-1\right)\left(n-2\right)x^{n-3}{y}^3}{\left(1\right)\left(2\right)}+\mathrm{...}+y^n$

The coefficients ${}_n{\text{C}}_k$ can be written as ${}_n{\text{C}}_k=\frac{n!}{k!\left(n-k\right)!}$ .

Let the 5^th term of the binomial be $z$ . So put $n=10$ , $k=4$ and solve as $z$ follows:

  $\begin{array}{c}z=\frac{10!}{4!\left(10-4\right)!}\cdot \left(x^{10-4}\right)\cdot {\left(-2\right)}^4\\ =210x^6\left(16\right)\\ =3360x^6\end{array}$

Thus the 5^th term of the binomial is $3360x^6$ .