# A Twisted Binomial Expansion!

Algebra Level 4

$\large{f(a,\ b,\ n) = \sum_{r=0}^n {n \choose r} \log \left(a^{n-r} b^r \right) }$

Let the above function satisfies for all positive integers $$a,b,n$$. If the value of $$f(7, \ 11, \ 21)$$ can be expressed as:

$\large{\left(P_1^A \cdot P_2^B \cdot P_3^C \right) \cdot \log(D)}$

for positive integers $$A,B,C,D$$ and distinct primes $$P_1, P_2, P_3$$ where $$D$$ isn't any perfect $$m^{th}$$ power of an integer with $$m>1$$, find the value of $$P_1 + P_2 + P_3 + A+B+C+D$$?

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