Question

The coefficient of $x^{15}$ in the product
$(1-x)(1-2x)(1-2^2.x)(1-2^3.x)...(1-2^{15}.x)$
is equal to

• A. $2^{105}-2^{121}$
• B. $2^{121}-2^{105}$
• C. $2^{120}-2^{104}$
• D. none of these

Answer 1

The correct option is A $2^{105}-2^{121}$

Coefficient of $x^{15}$ in given expression is given by

Multiply all $x$ terms with $1$, we get ${-2}^{120}$, next multiply with $-x$ leaving the term with $-2x,2^{2}x,2^{3}x$ respectively so

on, till $2^{15}x$.

We get,
${-2}^{120}{-2}^{119}{-2}^{118}...........-2^{105}$

Sum of all such terms will be $-2^{120}\left(\frac{1}{2^0}+\frac{1}{2^1}+\frac{1}{2^2}+...+\frac{1}{2^{15}}\right)$

$=-2^{105}(2^{15}+2^{14}+......+1)=-2^{105}\left(\frac{1-2^{16}}{1-2}\right)=2^{105}(1-2^{16})=2^{105}-2^{121}$