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• A-Level Maths

# The Binomial Expansion

Pascal's Triangle is used to expand binomial expressions like (a+b)ⁿ. It is created by summing adjacent pairs to find the number beneath this pair, starting from 1. Here are the first five rows:

The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a+b)ⁿ

### Factorial Notation

Parts of Pascal's triangle can be calculated quickly using factorial notation, ⁿCr (spoken "n choose r"):

## Expanding (a+b)ⁿ

When n ∈ ℕ (when n is a positive integer) the binomial expansion is in its simplest form:

The general term in an expansion of (a+b)ⁿ is given as:

## Expanding (1+x)ⁿ

If n is a fraction or a negative number, you need to use this form of the binomial expansion: