Pascal Triangle

You might be knowing the expansion of  (a + b)². Expansion of   (a + b)² = a² + 2ab + b². 

Similarly, (a + b)³ = a³ + 3a²b + 3ab² + b³.
If someone asks you to find the expansion of (a + b)¹⁰ or (a + b)¹⁵?

But expansions involving such large numbers are not difficult to write as you might be thinking. You can easily expand these types of expressions, if you know Pascal’s triangle.

Pascal’s triangle is a triangular array of the binomial coefficients. It is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (present day Iran) and China.

Now, you might be thinking how to write or remember this triangle. You can write Pascal’s by using these steps:

• At the top centre of your paper write the number “1”.
• On the next row write two 1’s, forming a triangle.
• On each subsequent row start and end with 1’s and compute each interior term by summing the two numbers above it.

Now, I will tell you how to use Pascal’s triangle.

To find (a + b)³, check the numbers in row whose second number is 3.

In this row, the numbers are 1, 3, 3 and 1, therefore, (a + b)³ = a³ + 3a²b + 3ab² + b³
To find (a + b)⁵, check the numbers in row whose second number is 5.

Numbers in this row are 1, 5, 10, 10, 5, 1, therefore, (a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

If it is (a – b)⁵, then + and – will come alternatively, (a – b)⁵ = a⁵ – 5a⁴b + 10a³b² – 10a²b³ + 5ab⁴ – b⁵
Isn’t it so simple to expand any binomial expansion with any power? Can you now tell what is the expansion of (a + b)¹⁰?