Mathematics - Binomial Theorem Question with Solution | TestHub

Using the binomial expansion ${\left(1+x\right)}^n={}^{n}C_0+{}^{n}C_{1}x+{}^{n}C_2{x}^2.......+{}^{n}C_n{x}^n$

We get,

$\alpha =\frac{\left(1+8\right){}^n-8n-1}{64}={}^{n}C_2+{}^{n}C_{3}8+{}^{n}C_4{8}^2+\dots$Similarly $\beta ={}^{n}C_2+{}^{n}C_{3}5+{}^{n}C_4{5}^2+\dots .$

So $\alpha -\beta =$${}^{n}C_3\left(8-5\right)+{}^{n}C_4\left(8^2-5^2\right)+\dots +{}^{n}C_n\left(8^{n-2}-5^{n-2}\right)$

So, option (C) is the answer.