Use the Binomial Theorem to expand the expression $(x+3{)}^6$ .

A) $x^6+729$
B) $x^6+3$
C) $x^6+3x^5+9x^4+27x^3+81x^2+243x+729$
D) $x^6+6x^5+15x^4+20x^3+15x^2+6x+729$
E) $x^6+18x^5+135x^4+540x^3+1215x^2+1458x+729$

The correct expansion of $(x+3{)}^6$ using the Binomial Theorem involves calculating the binomial coefficients (from Pascal's triangle or using combinations) and the powers of $x$ and $3$ accordingly. The coefficients for a binomial expansion of the form $(a+b{)}^n$ are given by $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ , where $n$ is the power and $k$ is the term number from $0$ to $n$ . The correct coefficients and terms for this expansion are found in option E, reflecting the accurate application of the Binomial Theorem.