I have always had a certain love for math and the neat things you can do with it. Here is a bit of information and shortcuts I have picked up in a few of my math classes.

Pascal's Triangle

Pascal's Triangle is a pretty neat thing. It is very simple to construct and can be used to understand a lot of different ideas. It follows a very simple form: start with a '1' and add the two digits above  to get the next number. The first few line look like this:

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5 10 10  5  1
1  6 15 20 15  6  1
...

The line numbers start at 0, and continue on ad infinitum. In order to generate this triangle, programmatically, you would use something like this:

vector<int> pascal(
      vector<int> prev, //the current (old) row data
      int *len,		//the length of the data
      int end,			//the row to retrieve
      int cur=0		//the current row we are on
      );

vector<int> pascal(vector<int> prev,int *len,int end,int cur) {

       //return immediately if we are at the last row
      if (cur==end) return prev;

     	//if the current vector length is 0, then set it to 1
      	if (*len==0) *len=1;

      	//create a temp vector (all 1's) to store the new row data
      	vector<int> t((*len)+1,1);

      	//sum the two rows
      	for(int i=1;i<(*len);i++)
            t[i]=prev[i-1]+prev[i];

      	//increase the length by 1
      	*len=(*len)+1;

      	//return the new row data
      	return pascal(t,len,end,cur+1);

}

Binomial Expansion

Remember binomials from algebra? They were the pair of numbers used to create or simplify polynomial expressions, something like:

You can use Pascal's triangle to find the coefficients of the polynomials. Let's begin by solving for the generic case:

See the coefficients so far, with $n = 2$ ? They are $[1 2 1]$, which corresponds to the second row in Pascal's triangle. But this could be a fluke, right, so let's jump ahead to $n = 5$ to see if that works as well.

[we know what $(a+b)^2$ is, so: ]

[note: notice that the coefficients of $(a+b)^4$ are (1 4 6 4 1) ! ]

There it is! The coefficients correspond to the rows on Pascal's Triangle!

Features

Now, to make things a little simpler, I will note some interesting "features" about what we just did.

General Formula for Binomial Expansion

The general formula for binomial expansion is:

Where $P_{ni}$ is the coefficient at row $n$ (starting from 0) and column $i$ in Pascal's Triangle. The formula means to add from $i=0$ to $n$ all the terms $(P_{ni}a^{n-i}b^i)$ , replacing $i$ with the number you are at. For example, supposing $i=3$, you would get:

since $P_3 = [1 3 3 1]$ , we finally get:

cleaning up a bit :

Exponents

Note in all the expansions, the first variable counts down from $n$ to $0$, while the second variable counts up from $0$ to $n$.

Does your binomial already have coefficients?

If your binomial already has coefficients, simply put them with their terms like so:

Using the commutative property, it can be rewritten as such:

Let's try an example!

Conclusion

As you can see, it is fairly easy to use Pascal's Triangle as a lookup table for binomial expansion's coefficients. I hope you have much more fun in your maths!