Alternating Coefficients

Algebra Level 4

Let $$P(x)=(x^{3}-x^{2}+x+4)^{2001}=a_{0}+a_{1}x+a_{2}x^{2}+....+a_{6003}x^{6003}$$.

The sum $$a_{0}+a_{2}+a_{4}+...+a_{6002}$$ can be expressed in the form $$\dfrac{5^{b}+c}{2}$$, where b and c are positive integers, and b is as large as possible. Find the value of $$5+b+c+2$$.

×