Chapter 12.1, Problem 1MRE

To determine

To determine the third term in the expansion of given binomial.

Answer to Problem 1MRE

Hence the third term of the expression is $12x^3{y}^2$

Explanation of Solution

Given information :

Given expression is ${\left(x+2y\right)}^5$

Binomial expansion of ${\left(a+b\right)}^n$ is given as,

  ${\left(a+b\right)}^n={}^n{C}_0{a}^n{b}^0+{}^{n-1}{C}_1{a}^{n-1}{b}^1+{}^{n-2}{C}_2{a}^{n-2}{b}^2\dots \dots {}^0{C}_n{a}^0{b}^n$

Hence the third term is ${}^{n-2}{C}_2{a}^{n-2}{b}^2$

We compare the given expression to general formula and we get the third term as,

  $\begin{array}{c}{}^3{C}_2{\left(x\right)}^3{\left(2y\right)}^2=\frac{3!}{2!\left(3-2\right)!}\left(x^3\right)\left(4y^2\right)\\ =12x^3{y}^2\end{array}$

Hence the third term of the expression is $12x^3{y}^2$