If coefficient of xn in (1+x)101(1-x+x2)100 is nonzero, then n can not be of the form

Question

If coefficient of xn in (1+x)101(1-x+x2)100 is nonzero, then n can not be of the form (A) 3 t+1 (B) 3 t (C) 3 t+2 (D) 4 t+1

Tardigrade Admin · Accepted Answer

(1+x)101(1-x+x2)100=(1+x)(1+x3)100 =(1+x)( C 0+C1 x3+C2 x6+ ldots+C100 x300) Clearly in this expression xλ will be present if λ=3 t, or λ=3 t+1 So, λ can not be of the form 3 t+2.

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

$3 t + 1$

$3 t$

$3 t + 2$

$4 t + 1$

$(1 + x)^{101} (1 - x + x^{2})^{100} = (1 + x) (1 + x^{3})^{100}$
$= (1 + x) (C_{0} + C_{1} x^{3} + C_{2} x^{6} + \dots + C_{100} x^{300})$
Clearly in this expression $x^{λ}$ will be present if $λ = 3 t$ ,
or $λ = 3 t + 1$
So, $λ$ can not be of the form $3 t + 2$ .

Q. If coefficient of xn in (1+x)101(1−x+x2)100 is nonzero, then n can not be of the form

3t+1

3t

3t+2

4t+1