Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the following expression $(bcd+acd+abd+abc)^7$.

Question

Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the following expression $(bcd+acd+abd+abc)^7$.
My attempt
I observe that $(bcd+acd+abd+abc)=cd(a+b)+ab(c+d)$ and this looks like the Vieta's relation for a quadratic equation. But I am not sure how to proceed. I want to avoid a bashing solution. Can any one help. Thanks in advance.

Answer 1

Hint: In order to get a term of $a^5b^5c^5d^6$ when you expand the product, how many each of $bcd$, $acd$, $abd$, and $abc$ do you need to choose? You might find this easier to think about if you think of each of them as adding one copy of each variable except for one. Or to put it another way, you can think of the product as $((abcd)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}))^7$.

Answer 2

Hint. Note that $$(bcd+acd+abd+abc)^7=(abcd)^7\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)^7$$ Then use the multinomial theorem.