The coefficient of x^256 in the expansion of (1-x)^101(x^2+x+1)^100 is

Question

The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}(x^2+x+1)^{100}$ is

Accepted Answer

$$(1-x)^{101}(1+x+x^2)^{100} = (1-x)\left[(1-x)(1+x+x^2)\right]^{100}=(1-x)(1-x^3)^{100}$$

$$=(1-x)\sum_{r=0}^{100}(-1)^r\binom{100}{r}x^{3r}$$

For $x^{256}$: from $-x\cdot(-1)^r\binom{100}{r}x^{3r}$ term, $3r+1=256\Rightarrow r=85$.

Coefficient $=-(-1)^{85}\binom{100}{85}=\binom{100}{85}=\binom{100}{15}$

Solutions · 1