Consider the polynomial

P(y) = (y+1)(y² + 2)(y⁴ + 4)(y⁸ + 8)...(y¹⁰²⁴ + 1024)

Determine the value of A, given that the coefficient of y²⁰¹² in the expansion of P(y) is equal to 2^A

Product[r=0,…10  of  (y^2^r + 2^r)].

Since each bracket contains y raised to a different power of 2, the

binary representation, 2012 = 11111011100₂, tells us that y²⁰¹² derives

from the product y^(2²+2³+2⁴+2⁶+2⁷+2⁸+2⁹+2¹⁰), these particular

powers coming uniquely from 8 of the 11 brackets. The coefficient of

y²⁰¹² will therefore come from the powers of 2 in the ‘missing’ 3 brackets.

Thus:    A = coefficient of y²⁰¹² = 2^(0 + 1 + 5)  =  64.

Posted by Harry on 2014-08-10 16:53:40