The Perplexing Polynomial

This is not an original problem

Let p(x) be the polynomial $(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k$ , where a, b, ..., k are integers. When expanded in powers of x, the coefficient of x is -2 and the coefficients of ${ x }^{ 2 },{ x }^{ 3 },...,{ x }^{ 32 }$ are all zero.

Find k. Then, write k in the form ${ 2 }^{ n }-2^{ m }$ What is n+m?

based on a problem from the USAMO