polynomials – How to prove that a coefficient is divisible by two using Multinomial theorem?

Given the product of terms $ (a + b + 2*c + 2*d)*(2*a + 2*b + c + d)*(a + b + c + 2*d)*(a + 2*b + c + d) $ prove that coefficient in combination $a*b*c*d$ is not divisible by two.

Multinomial theorem defines the coefficient formulas for the $ (a+b+c+d)^n $ but they can’t be applied to the task above.

How to solve this task in general case without polynomial expansion ?