Use the Binomial Theorem to expand the expression $(5a+4b{)}^4$ .

A) $625a^4+256b^4$
B) $a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$
C) $625a^4+500a^{3}b+400a^2{b}^2+320a{b}^3+256b^4$
D) $a^4+16a^{3}b+96a^2{b}^2+256a{b}^3+256b^4$
E) $625a^4+2000a^{3}b+2400a^2{b}^2+1280a{b}^3+256b^4$

The correct expansion using the Binomial Theorem involves calculating the coefficients using the binomial coefficients (which in this case are 1, 4, 6, 4, 1 for the powers of 4 down to 0) and raising both $a$ and $b$ , as well as their coefficients (5 and 4, respectively), to the appropriate powers. The correct coefficients for each term are obtained by multiplying the binomial coefficient by the respective powers of 5 and 4, leading to the expansion provided in option E.