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 \(5+b+c+2\).