#Combinatorics #Year11 #Ext1 >[!info]- [The binomial theorem | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/n11/fa00679960) >- ME1-11-05 uses the binomial theorem to solve problems and prove identities ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | --------------------------------- | --------------------- | --------------------------------- | | [[Algebraic Techniques C]] | - expanding binomials | - binomial expansion of any power | | [[Permutations and Combinations]] | - combinations | - binomial theorem | ## The binomial theorem - Recognise that a binomial is an expression with two terms, and that a binomial expansion is an expansion of a power of a binomial - Examine the symmetry formed by the coefficients of decreasing powers of $x$ in the expansion of $(x+y)^n$ for $n=0,1,2,3,4,5$ and arrange the coefficients into Pascal's triangle - Recognise the equivalence between the coefficient of $x^{n-r}y^r$ in the expansion of $(x+y)^n$ and $^nC_r$, when $n$ is a positive integer - Use patterns and symmetry in Pascal's triangle to confirm the identities $^nC_r=^{n-1}C_{r-1}+^{n-1}C_r$ for $1 \le r \le n-1$ and $^nC_r=^nC_{n-r}$ for $0 \le r \le n$ - Derive the binomial theorem: $(x+y)^n=^nC_0x^n+^nC_1x^{n-1}y+^nC_2x^{n-2}y^2+\ldots+^nC_{n-1}xy^{n-1}+^nC_ny^n$, when $n$ is a positive integer - Apply the binomial theorem to expand and simplify expressions of the form $(x+y)^n$ - Use the binomial theorem to determine the coefficient of a term with a specific power or the constant term in a binomial expansion - Use the identities $^nC_0=1$, $^nC_n=1$, $^nC_r=^{n-1}C_{r-1}+^{n-1}C_r$ for $1 \le r \le n-1$ and $^nC_r=^nC_{n-r}$ for $0 \le r \le n$ to simplify expressions involving the binomial coefficients - Prove identities involving binomial coefficients in binomial expansions by substituting values, comparing coefficients or applying a combinatorial argument to a specified context - Apply given or proven identities involving binomial coefficients to prove further identities, without the use of calculus