THE PASCAL'S TRIANGLE AND BINOMIAL THEOREM

BINOMIAL THEOREM AND THE PASCAL'S TRIANGLE

We have all learned to expand binomials with different powers using the factorial method: 

Here we have to calculate the coefficients by calculating the combinations :

 

using the calculator or the formula given below : 

Using this method, we could keep on going but eventually, the exponents would get so large that it would become difficult to simplify them. 

Thus, we need a better way, and luckily a 17th-century French mathematician has already found one.

Blaise Pascal  found a numerical pattern, called Pascal's Triangle, for quickly expanding a binomial.

Here's what the Pascal Triangle looks like : 

The magic that it does is that the rows of  this triangle show the coefficients of different powers of binomials :

For example: 

• The 1st row shows the coefficient of the expansion (a + b)0 = 1
• The 2nd row shows the coefficients of the expansion (a + b)1 = 1a + 1b
• The 3rd row shows the coefficents of the expansion (a + b)2 = 1a2 + 2ab + 1b2 
• The 4th row shows the coefficients of the expansion (a + b)3 = 1a3 + 3a2b + 3ab2 + 1b³

And it goes on.

The interesting thing is that you do not need to learn the Pascal's triangle. If you notice, it is a sequence and you can simply go on expanding the triangle for as many rows as you want : 

• The first row has 1.
• For the next row you simply add the above number with its left and right number; as in the first row its zero on both sides, the second row gets 1 and 1
• In the third row we write 1, then add the second row's 1 and 2 to get 3 twice, and again a 1.

This can be better understood by the diagram below: 

Hence, it shows how the Pascal's triangle can be used to work out Binomial expansions.

The Pascal's triangle has many applications as well as many hidden wonders in itself (some of which have been discovered while some are yet to be explored), and they'll be discussed in detail in later articles.