<s>[INST] The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x)\cdots (1-2^{15}x)$ is .... Problem : 
The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x)\cdots (1-2^{15}x)$ is 
(a) $2^{105}-2^{121}$ 
(b)  $2^{121}-2^{105}$ 
(c) $2^{120}-2^{104}$ 
(d) $2^{110}-2^{108}$
My approach : 
Tried to find pattern if any in the given expansion, so expanded first three terms : 
$(1-x)(1-2x)(1-2^2x) = (1-3x+2x^2)(1-4x) = (1-3x+2x^2 -4x+12x^2 -8x^3)$
$= (1-7x+14x^2 -8x^3)$
But didn't find any pattern of this series please suggest how to proceed will be of great help thanks. 
 [/INST] I'd rescale as $$2^0\cdot2^1\cdot2^2\cdots2^{15}\left[(1-x)\left(\frac12-x\right)\left(\frac14-x\right)\cdots\left(\frac1{2^{15}}-x\right)\right]$$ The coefficient of $x^{15}$ in the polynomial within the brackets is the negative of the sum of the roots (since there are $16$ linear factors). So the answer is $$-2^0\cdot2^1\cdot2^2\cdots2^{15}\left[1+\frac12+\frac14+\cdots+\frac1{2^{15}}\right]=-2^{120}\left[2-\frac{1}{2^{15}}\right]=2^{105}-2^{121}$$
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