Dear Maths Help, I can't understand what Pascal's Triangle has got to do with multiplying out brackets. |
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 + x^{2} | |
(1 + x)^{3} | = | 1 + 3x + 3x^{2} + x^{3} |
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 n^{th} row of Pascal's triangle. |
You don't have to write out Pascal's triangle to find out how brackets expand.
The r^{th} item (counting from 0) in the n^{th} row of Pascal's triangle is
You may have this button on your calculator. Or Maths Help can show you.