Coefficient Of Your Knowledge!

Let $f=abcd, A=1/a, B=1/b, C=1/c, D=1/d$ . Substituting these values into the expression simplifying it into the form $(A+B+C+D)^7f^7$ . This new expression now changes the value $a^5b^5c^5d^6 = A^2B^2C^2Df^7$ .

Finding the coefficient we're looking for can be done by finding the number of possible arrangements of $AABBCCD$ exist. A quick count done by filling each term in one at a time gives $\frac{7!}{2!2!2!1!}=630$ .