Binomial expansion

The nCr function

The function nCr is used in the binomial expansion and in the binomial distribution.

In this tutorial you are shown how to work it out manually and on a calculator. The example uses a Casio fx-series calculator.

{^n\mathrm{C}_r} \text{ formula:}
  • {^n\mathrm{C}_r} = \dfrac{n!}{(n-r)! r!}

Binomial Expansion using the nCr method

In this video tutorial you are introduced to the binomial expansion as a method which reduces the amount of working in expanding a bracket to a given positive power.

Binomial Expansion for a positive integer n:

  • (a+b)^n \equiv {^n\mathrm{C}_0} a^n b^0 + {^n\mathrm{C}_1} a^{(n-1)} b^1 + {^n\mathrm{C}_2} a^{(n-2)} b^2 + ... + {^n\mathrm{C}_n} a^0 b^n

Finding particular terms using the nCr method

In this tutorial you are shown how to find a particular term or coefficient of a term in a binomial expansion without having to resort to the full expansion.

The following examples are used

Examples in the video

  1. \text{Find the term in } x^5 \text{ in the expansion } \left(5-2x\right)^8
  2. \text{Find the coefficient of } \dfrac{1}{x^3} \text{ in the expansion } \left(2 - \dfrac{3}{x} \right)^7

Binomial Expansion using Pascal’s Triangle

Pascal’s triangle is a pattern of numbers which can be used to work out values of the binomial coefficients nCr as the following tutorial explains.

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