How does Pascal’s Triangle relate to binomial expansion?

Hint: We can expand a particular given series in many ways. We can use normal binomial expansion or we can use Pascal’s Triangle method to find the expansion of any given series. To solve a series using the Pascal’s Triangle method, we firstly need to have a complete idea about how, where and when we can use this particular method for expansions. Pascal’s Triangle is a triangular array that is constructed by adding the adjacent terms in the previous row.

Complete step-by-step solution:
Now we start off with the solution to the problem by writing off Pascal’s Triangle for the first few terms. They are written as,
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1st Row} \\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{2nd Row} \\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{3rd Row} \\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{4th Row} \\
&\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\text{5th Row} \\
& 1\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\text{6th Row} \\
\end{align}\]
So in any given problem in which we need to find out the binomial expansion of any series or equation we need to write this Pascal’s Triangle and then from this we can find the complete expansion of the series by writing the unknown parameters along with one of the rows of the triangle. Let us consider an example for the binomial expansion of \[{{\left( x+y \right)}^{5}}\] . Here we need to consider the \[{{\text{6}}^{\text{th}}}\] row of the Pascal’s Triangle and then we can write that,
\[{{\left( x+y \right)}^{5}}={{x}^{5}}+5{{x}^{4}}y+10{{x}^{3}}{{y}^{2}}+10{{x}^{2}}{{y}^{3}}+5x{{y}^{4}}+{{y}^{5}}\]
So in this way we can easily find the expansion using the Pascal’s Triangle method.

Note: For solving expansion problems using the Pascal’s Triangle method we need to have a clear idea of what the method is and how to implement the same. An important thing that we must always remember is that, when we want to find the value of the expansion of the ${{n}^{th}}$ term then we must consider the ${{\left( n+1 \right)}^{th}}$ row of the Pascal’s Triangle to find the equivalent series. We can also solve a problem using normal binomial expansion which we are accustomed to, however the implementation of Pascal’s Triangle is much simpler.