Question

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

• A. $3t+1$
• B. $3t$
• C. $3t+2$
• D. $4t+1$

Answer 1

The correct option is C $3t+2$

${(1+x)}^{101}{(1-x+x^2)}^{100}=(1+x){(1+x^3)}^{100}=(1+x)({}^n{C}_0{+}^n{C}_1{x}^3{+}^n{C}_2{x}^6+...{+}^n{C}_{100}{x}^{300})$

Clearly, in the expression $x^{\lambda }$ will be present if $\lambda =3$, or $\lambda =3t+1$

So, $\lambda$ cannot be of the form $3t+2$