Vector Spaces & Bases · Superchallenge problem

In $P_4$ (polynomials of degree at most $4$) let $B = \{1, x, x^2, x^3, x^4\}$ and let $C$ be the basis of falling factorials $\{1,\ x,\ x(x-1),\ x(x-1)(x-2),\ x(x-1)(x-2)(x-3)\}$. Find the sum of all $25$ entries of the change-of-basis matrix $P_{C\leftarrow B}$, whose columns are the $C$-coordinate vectors of $1, x, x^2, x^3, x^4$.