Mathematics - Binomial Theorem Question with Solution | TestHub

$$\begin{array}{r}(x+a{)}^n={}^n{C}_0{x}^n+{}^n{C}_1{x}^{n-1}a+{}^n{C}_2{x}^{n-2}{a}^2\\ +{}^n{C}_3{x}^{n-3}{a}^3+{}^n{C}_4{x}^{n-4}{a}^4+\dots .\\ =\left({}^n{C}_0{x}^n+{}^n{C}_2{x}^{n-2}{a}^2+{}^n{C}_4{x}^{n-4}{a}^4+\dots .\right)+\\ +\left({}^n{C}_1{x}^{n-1}a+{}^n{C}_3{x}^{n-3}{a}^3+{}^n{C}_5{x}^{n-5}{a}^5\right)+\dots .\end{array}$$ $=A+B$ ....(i) Similarly, $(x-a{)}^n=A-B$ ....(ii) Multiplying eqns. (1) and (2), we get $${\left({\mathrm{x}}^2-{\mathrm{a}}^2\right)}^{\mathrm{n}}={\mathrm{A}}^2-{\mathrm{B}}^2$$