Mathematics - Binomial Theorem Question with Solution | TestHub

$(1+x)\left(1+x+x^2\right)\dots .\left(1+x+\dots +x^n\right)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots$. Put, $\mathrm{x}=1$, $2\times 3\times 4\times \dots (n+1)=a_0+a_1+a_2+a_3+\dots .$ Put $\mathrm{x}=-1$, $\begin{array}{rlr}& 0=a_0-a_1+a_2-a_3+\dots & (n+1)!=2\left[a_0+a_2+\dots \right]\Rightarrow a_0+a_2+\dots =\frac{(n+1)!}{2}\end{array}$