If a, b and c are positive real numbers such that$\left\{\right(1+a\left)\right(1+b\left)\right(1+c){\}}^7>k^m\left(a^4{b}^4{c}^4\right)$, find the value of $k + m.$

We have, 

$\begin{array}{l}(1+a)(1+b)(1+c)\\ =1+(a+b+c)+(ab+bc+ca)+abc\end{array}$

As we know that $\mathrm{AM}\ge \mathrm{GM}$

$\begin{array}{l}\frac{(a+b+c+ab+bc+ca+abc)}{7}\ge \sqrt[7]{a^4{b}^4{c}^4}\\ \Rightarrow (a+b+c+ab+bc+ca+abc)\ge 7\sqrt[7]{a^4{b}^4{c}^4}\\ \Rightarrow (1+a+b+c+ab+bc+ca+abc)\\ >(a+b+c+ab+bc+ca+abc)\ge 7\sqrt[7]{a^4{b}^4{c}^4}\end{array}$

$\begin{array}{l}\Rightarrow (1+a)(1+b)(1+c)\ge 7\sqrt[7]{a^4{b}^4{c}^4}\\ \Rightarrow (1+a{)}^7(1+b{)}^7(1+c{)}^7\ge 7^7\left(a^4{b}^4{c}^4\right)\\ \text{ Thus }k=7\text{ and }m=7\\ \text{ Hence, the value of }k+m\text{ is }14\text{. }\end{array}$