Please tell above answer quickly and as soon as possible

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The coefficient of x¹⁵ in (1 – x)(1 – 2x)(1 – 2²x) … (1 – 2¹⁵x) means the product of coefficients of x of any 15 factors of the above expression i.e. we need to multiply coefficients of any 15 factors i.e. leaving one factor of total 16 factors at a time.

Coefficient of x¹⁵ is coefficient of x in  (1 – x)(1 – 2x)(1 – 2²x) … (1 – 2¹⁵x) / (1 – x) +

  (1 – x)(1 – 2x)(1 – 2²x) … (1 – 2¹⁵x) / (1 – 2x)  +  (1 – x)(1 – 2x)(1 – 2²x) … (1 – 2¹⁵x) / (1 – 2²x)  +

  …. + (1 – x)(1 – 2x)(1 – 2²) … (1 – 2¹⁵x) / (1 – 2¹⁵x)

coefficient of x is (–1)(–2)(–2²)...(–2¹⁵) / (–1)  + (–1)(–2)(–2²)...(–2¹⁵) / (–2) + (–1)(–2)(–2²)...(–2¹⁵) / (–2²)

            + …. + (–1)(–2)(–2²)...(–2¹⁵) / (–2¹⁵)

   =  – (2^1+2+3+...+15) – (2^1+2+3+...+15) / 2¹ – (2^1+2+3+...+15) / 2² – …. – (2^1+2+3+...+15) / 2¹⁵

   =  – (2^1+2+3+...+15) (1 + 1/2 + 1/2² + … + 1/2¹⁵)

   =  – (2¹²⁰) (1 – (1/2)¹⁶) / (1 – 1/2)  =  – (2¹²⁰) (1 – 1/2¹⁶) / (1/2)  =  – (2¹²¹) (1 – 1/2¹⁶)

   = – 2¹²¹  +  2^{121 – 16}  =  – 2¹²¹  +  2¹⁰⁵ = required coefficient of x¹⁵ .

Here note that 1 + 1/2 + 1/2² + … + 1/2¹⁵ is a GP whose first term is 1, common ratio is 1/2 and

number of terms is 16.