Use the Binomial Theorem to expand the expression $(x+3y{)}^4$ .

A) $x^4+12x^{3}y+54x^2{y}^2+108x{y}^3+81y^4$
B) $x^4+3x^{3}y+9x^2{y}^2+27x{y}^3+81y^4$
C) $x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4$
D) $x^4+81y^4$
E) $x^4+15x^{3}y+90x^2{y}^2+135x{y}^3+81y^4$

The correct expansion using the Binomial Theorem involves calculating the coefficients using the binomial coefficients formula, which for $(x+3y{)}^4$ gives coefficients of 1, 4, 6, 4, 1, and multiplying these by the appropriate powers of 3y, resulting in the terms $x^4$ , $12x^{3}y$ , $54x^2{y}^2$ , $108x{y}^3$ , and $81y^4$ .