How much input maps to output?

[tickle_monste]

Homework Statement

I have polynomials of the form:
x₁ + x₂ + x₃ + x₁x₂ + x₁x₃ + x₂x₃ + x₁x₂x₃

The x's can be a combination of 1s and 0s, so all the possible inputs could be written as strings: 000, 100, 010, 001, 110, 101, 011, 111

The polynomials can also have coefficients a...g, so could just be written abcdefg

My question is, given a polynomial abcdefg, how many of my possible inputs will be mapped to a given output?

Homework Equations

The Attempt at a Solution

No idea where to start. I'm familiar with the concept of a Lebesgue integral, but have no idea how to actually do a Lebesgue integral, let alone discretize it... for 3 variables... or if this is even the line of thought I should be going down. I have put in a lot of thought to this but have not had even enough success to have made even a serious attempt at a solution, as I don't know where to begin. I appreciate any help anyone can give me on this problem.

 

[micromass]

So, let's start with the following. Given abcdefg, calculate all possible outputs...

 

[tickle_monste]

Given a polynomial abcdefg, there are 8 possible inputs and thus 8 possible outputs, and I have 8 equations describing each possible output. They are:

p(000) = 0
p(100) = a
p(010) = b
p(001) = c
p(110) = a+b+d
p(101) = a+c+e
p(011) = b+c+f
p(111) = a+b+c+d+e+f+g

Given a polynomial abcdefg and an integer k, I need to find the number of the above expressions which will evaluate to k.