Application to the Binomial Theorem

The binomial theorem is a way of expanding any binomial expression to some nth power.

The coefficients from the binomial theorem make the entries of each row of Pascal's Triangle. Where n=0 corresponds to the 1st row, n=1 corresponds to the 2nd row, and so on.


For any skeptics of the binomial theorem, here is a simple proof of why it works: Proof of the Binomial Theorem


The Binomial Theorem can also help prove some of the properties of Pascal's Triangle. It is easy to look at Properties of Pascal's Triangle and be amazed by it, and then just move on. But these properties don't just magically occur, all of them can be proved as to why they happen. So, instead of being amazed that these properties exist and then just moving on, let's be amazed as to why these properties exist. The following are proofs of some of the properties of Pascal's Triangle:

As a reminder, in the video found in "Properties", the sum of any nth row of Pascal's Triangle is equal to 2ⁿ. Thus, all the powers of 2 can be found in Pascal's Triangle.



Earlier, it was stated that the coefficients of the Binomial Theorem can be used to make Pascal's Triangle. This is the same as setting a=b=1 in the binomial theorem, which gives:



It can be seen that the terms of this expansion will correspond to the elements of the n^th row of Pascal's Triangle.
It is then trivial to see that (1+1)ⁿ = 2ⁿ. Thus, the sum of the elements of any row, n, is equal to the sum of the terms of the binomial expansion when a=b=1 which is 2ⁿ.

The Binomial Theorem can also be used to prove why the square of any of the numbers in the second row (which contains the natural numbers) can be found by the sum of the numbers in the third row that are adjacent to the number in third row.



Since one of the adjacent numbers is always in the same row, n, it can be denoted as the third coefficient of any binomial expansian which is always _nC₂. The other number is always in the row below the nth row, it can be denoted as the third coefficient in the binomial expansion for n+1 which is _n+1C₂. Thus, the squares property found in Pascal's Triangle can be proved by proving that n²=_nC₂+_n+1C₂:

The Hockey Stick Identity can be explained mathematically through the following identity,

The following shows an animated explanation of this identity:

This does not have anything to do with how the binomial theorem applies to Pascal's Triangle but as mentioned in the video found in the "Properties" tab, mathematicians have created a theorem for polynomials larger than 2 terms called the multinomial theorem. For anyone that is curious this is what the multinomial theorem looks like:

When k=2 the coefficient would be:

Which is the coefficient for the binomial theorem. For more information about the multinomial theorem and even a proof of it, click on the following link:
More on the Multinomial Theorem