Question

Using binomial theorem, evaluate each of the following:
$(102{)}^5$

Answer 1

$\begin{array}{c}{\left(102\right)}^5\\ {\left(102\right)}^5={\left(100+2\right)}^5\\ ={}^5{C}_0{\left(100\right)}^5{2}^0+{}^5{C}_1{\left(100\right)}^4{2}^1+{}^5{C}_2{100}^3{2}^2\\ {+}^5{C}_3{100}^2{2}^3+{}^5{C}_4{100}^1{2}^{4}6{+}^5{C}_5{100}^0{2}^5\\ ={100}^5+5{\left(100\right)}^4\times 2+10{\left(100\right)}^{3}4+10{\left(100\right)}^{2}8\\ +5\left(100\right)16+32\\ ={10}^{10}+{10}^0+4\times {10}^9+8\times {10}^5+8\times {10}^3+32\\ =11040808032\end{array}$