# 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 ?