Question

If the coefficient of $x^n$ in $(1+x{)}^{101}(1-x+x^2{)}^{100}$ is nonzero, then $n$ cannot be of the form

• A. $3r+1$
• B. $3r$
• C. $3r+2$
• D. none of these

Answer 1

The correct option is C $3r+2$

This can be re-written as
$(1+x)\left[\right(1+x\left)\right(1-x+x^2){]}^{100}$
$=(1+x)(1+x^3{)}^{100}$
$=(1+x^3{)}^{100}+x(1+x^3{)}^{100}$
Hence coefficient of $x^n$ will be
$={}^{100}{C}_{\frac{n}{3}}+{}^{100}{C}_{\frac{n-1}{3}}$ ...(i)
Hence from i, we can say the n can of type , $3r$ or $3r+1$ but not of type $3r+2$.
Where $rϵW$