The coefficient of x256 in the expansion of (1 - x)101 (x2 + x + 1)100 is :

Question

The coefficient of x256 in the expansion of

(1 - x)101 (x2 + x + 1)100 is :

Accepted Answer

${(1 - x)^{101}}{({x^2} + x + 1)^{100}}$

Coefficient of $${x^{256}} = {[(1 - x)(1 + x + {x^2})]^{100}}(1 - x) = {(1 - {x^3})^{100}}(1 - x)$$

$$ \Rightarrow ({}^{100}{C_0} - {}^{100}{C_1}{x^3} + {}^{100}{C_2}{x^6} - {}^{100}{C_3}{x^9}...)(1 - x)$$

$$\sum {{{( - 1)}^r}{}^{100}{C_r}{x^{3r}}(1 - x)} $$

$$ \Rightarrow 3r = 256$$ or $$255 \Rightarrow r = {{256} \over 3}$$ (Reject)

r = 85

Coefficient = ${}^{100}{C_{85}} = {}^{100}{C_{15}}$

Solutions · 1