 # Binomial Theorem

 From: Tony Mowlem Date: 20 February 1999 Subject: Pascal's Triangle and Expanding Brackets Dear Maths Help, I can't understand what Pascal's Triangle has got to do with multiplying out brackets. Please explain.

### Maths Help suggests:

For readers who are not familiar with it, Pascal's Triangle (first five rows) is shown below:

 Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1

Each number in the table is the sum of the two numbers above it.

Now consider what happens if you expand (multiply out) these brackets:

 (1 + x)0 = 1 (Anything to power 0 is 1) (1 + x)1 = 1 + x (1 + x)2 = 1 + 2x + x2 (1 + x)3 = 1 + 3x + 3x2 + x3

What you will notice is that:

 In the expansion of (1 + x)n, the coefficients of the powers of x follow the same pattern as the nth row of Pascal's triangle.

## Time-Saving Tip

You don't have to write out Pascal's triangle to find out how brackets expand.

The rth item (counting from 0) in the nth row of Pascal's triangle is You may have this button on your calculator. Or Maths Help can show you.

Tony asked a follow-up question about this. Click to see that question and answer.
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