The Binomial Theorem

combinatorics year11 ext1

The binomial theorem | NSW Curriculum Website

• ME1-11-05 uses the binomial theorem to solve problems and prove identities

📖 Prior Knowledge

Content	Prerequisite relationships
Algebraic Techniques A	- Expand binomial products → Construct Pascal’s triangle
Indices A	- Apply index laws to algebraic terms → Find a specific term in an expansion
Permutations and Combinations	- Count combinations using ${}^n{C}_r$ → Relate Pascal’s triangle to ${}^n{C}_r$ - Identities for combinations → Simplify binomial coefficient expressions

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 ${}^n{C}_r$, when $n$ is a positive integer
• Use patterns and symmetry in Pascal’s triangle to confirm the identities ${}^n{C}_r{=}^{n-1}{C}_{r-1}{+}^{n-1}{C}_r$ for $1\le r\le n-1$ and ${}^n{C}_r{=}^n{C}_{n-r}$ for $0\le r\le n$
• Derive the binomial theorem: $(x+y{)}^n{=}^n{C}_0{x}^n{+}^n{C}_1{x}^{n-1}y{+}^n{C}_2{x}^{n-2}{y}^2+\dots {+}^n{C}_{n-1}x{y}^{n-1}{+}^n{C}_n{y}^n$, 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 ${}^n{C}_0=1$, ${}^n{C}_n=1$, ${}^n{C}_r{=}^{n-1}{C}_{r-1}{+}^{n-1}{C}_r$ for $1\le r\le n-1$ and ${}^n{C}_r{=}^n{C}_{n-r}$ 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