Pascal’s Triangle and Binomial Expansion (Leaving Cert Mathematics): Revision Notes

Pascal's Triangle and Binomial Expansion

Pascal's Triangle is a triangular arrangement of numbers where:

• Each row begins and ends with the number one.
• Every other number in the row is the sum of the two numbers directly above it from the previous row.

It is used to:

• Find the coefficients in binomial expansion.
• Solve probability problems using combinations.
• Explore number patterns.

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Each row corresponds to the coefficients in the expansion of a binomial expression $(a+b)^n$. For example :

• $(a+b)^0=1$
• $(a+b)^1=a+b$
• $(a+b)^2=a^2+2ab+b^2$

For higher powers, there comes a point where multiplying out the expression becomes too tedious, so we can use binomial expansion.

Binomial expansion is a way of writing out expressions without multiplying them out in full step-by-step.

We get a desired coefficient using the formula :

Example

Example