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NSW Y11 Maths Standard NSW Y11 Maths Advanced NSW Y11 Maths Extension 1

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Functions Polynomials Inverse Trig Functions and Identities Rates of Change Permutations and Combinations

NSW Y11 Maths - Extension 1 Permutations and Combinations Binomial Theorem

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Binomial Theorem Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-A1.2: The binomial expansion and Pascal’s triangle. This will require student to 

  • expand \((𝑥 +𝑦)^𝑛\) for small positive integers \(𝑛\) (ACMMM046)
  • derive and use simple identities associated with Pascal’s triangle (ACMSM009)

Ref: https://educationstandards.nsw.edu.au/

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Videos

Videos relating to Binomial Theorem.

  • Binomial Theorem - Video - The Binomial Theorem

  • Binomial Theorem - Video - Binomial Theorem - General Formula

  • Binomial Theorem - Video - Binomial Theorem Find Term independent of variable x

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Theory

The coefficients of \({x^k}\) in the expansion of \({\left( {1 + x} \right)^n}\) can be written as \(^n{C_k}\)

For example:  \({\left( {1 + x} \right)^4} = {}^4{C_0} + {}^4{C_1}x + {}^4{C_2}{x^2} + {}^4{C_3}{x^3} + {}^4{C_4}{x^4}\)

Which is: \({\left( {1 + x} \right)^4} = 1 + 4x + 6{x^2} + 4{x^3} + {x^4}\)

In general: \({\left( {x + y} \right)^n} = {}^n{C_0}{x^n} + {}^n{C_1}{x^{n - 1}}y + {}^n{C_2}{x^{n - 2}}{y^2} + ... + {}^n{C_k}{x^{n - k}}{y^k} + ... + {}^n{C_n}{y^n}\) where \({T_{k + 1}} = {}^n{C_k}{x^{n - k}}{y^k}\) is called the general term.