Question

If the coefficient of $x^n$ in $(1+x{)}^{101}(1-x+x^2{)}^{100}$ is $b,$ then which of the following is/are correct ?
$(r\in W)$

• A. $b=0$ when $n=3r$
• B. $b\ne 0$ when $n=3r$
• C. $b\ne 0$ when $n=3r+1$
• D. $b=0$ when $n=3r+2$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $b\ne 0$ when $n=3r$
C $b\ne 0$ when $n=3r+1$
D $b=0$ when $n=3r+2$

We have,
$(1+x{)}^{101}(1-x+x^2{)}^{100}=(1+x)((1+x{)}^{100}(1-x+x^2{)}^{100})$
$=(1+x)(1+x^3{)}^{100}=(1+x)[C_0+C_1{x}^3+C_2{x}^6+\dots +C_{100}{x}^{300}]$
$=(1+x)\sum _{r=0}^n{}^n{C}_r{x}^{3r}=\sum _{r=0}^n{}^n{C}_r{x}^{3r}+\sum _{r=0}^n{}^n{C}_r{x}^{3r+1}$

We can observe that $x^{3r+2}$ terms are missing, which means the coefficients of $x^{3r+2}$ terms are $0.$
Similarly, coefficients of $x^{3r}$ and $x^{3r+1}$ terms be non-zero.