Perhaps the most interesting relationship found in Pascal’s Triangle is how we can use it to find the combinatorial numbers (see image above). Recall the combinatorics formula *n* choose *k* (if you’re blanking on what I’m talking about check out this post for a review). We find that in each row of Pascal’s Triangle *n* is the row number and *k* is the entry in that row, when counting from zero.^{1}

## Definition

For anyone who has experience with Mathematics, Pascal’s Triangle should need no introduction. It was discovered by Chinese mathematician Jia Xian, when he discovered a triangle that was a representation of the coefficients in a binomial expansion. This triangle was further popularized by Chinese mathematician Yang Hui, which is why it is often known as the Yanghui triangle in China. Despite this, the modern world mostly knows this triangle as Pascal’s Triangle, after 17th century French mathematician Blaise Pascal. ^{2}

## Who

In the real world, Pascal’s Triangle translates into the complex topic of graph theory who’ve includes turning mapping information into data, finding shortest paths, airplane rules and airport control, computer graphics, engineering, data management, search algorithms and more. Pascal’s Triangle is also often used in architecture and design, because since buildings are 3-D, using Pascal’s Triangle in expansions and measurements is common. ^{3}

What are some real world examples of the use of the binomial theorem? ^{4}

- Binomial theorem is heavily used in probability theory, and a very large part of the US economy depends on probabilistic analyses. It is most useful in our economy to find the chances of profit and loss which is a great deal with developing economy.
- Binomial theorem and distribution is used in higher mathematics and calculation. Suppose you have given an equation with have degree 28 i.e like 2*x^28 +3*x^27+……..+3. You can find here all 28 roots of x In certain scientific research binomial is very helpful to solve impossible equations.if you have seen Einstein equations there is a lot use of binomial theorem. That is why we have now very great theories and laws by Sir Albert Einstein.
- Moreover binomial theorem is used in forecast services .the future weather forecasting is impossible without binomial theorem.the disaster forecast is also depend upon binomial theorems.
- The selection is the most using application in our life. Popularly known it uses this theorem to give ranks to the candidates.The probability will be impossible without binomial distribution.
- It is used in architecture in giving shape and determining the areas of infrastructure to find about the amount of material to be use in that. it help in the estimation and to find the total expenditure to build the building .so all financial sites depend indirectly on binomial theorem.

## What

Many of you already know about Pascal’s triangle. Pascal’s triangle is a triangular arrangement of number that’s really useful in the expansion of any **binomial expression**, such as *(x+y)^n*. Not only in algebra, Pascal’s triangle also used in **set theory**, **probability theory** and **combinatorics**. ^{5}

**How do I solve (√a+√b) ^8 with pascal’s triangle?**^{ 6}

(We don’t “solve” these, we “expand” them.)

I feel that there is no need to use the old traditional “formula method” or Pascal’s triangle for finding binomial expansions. I much prefer the following approach.

Let’s experiment by looking at a few expansions.

Then, just looking at the NUMBERS (coefficients), we get Pascal’s Triangle:

Many years ago, I read that our old friend, Newton, saw a simple pattern for producing these **coefficients** without having to use Pascal’s triangle as follows:

This is the simple pattern I would use!

I call this the “** thinking method**” as opposed to the

**”.**

*“formula method*I think it would be very instructive and helpful to examine how I have expanded the following without resorting to using some standard general term formula.

**How can you memorize Pascal’s Triangle?** ^{7}

It is not a good idea to memorise Pascal’s Triangle!

Suppose you need a few terms of something like…

It is a much better idea to remember a **pattern** which will work for any sized number.

The pattern is much easier to remember than the general term or Pascal’s triangle!

Now just see how easy it is to expand a few terms of…

## Why

See Theoretical Knowledge Vs Practical Application.

## How

Many of the **References** and **Additional Reading** websites and **Videos** will assist you with understating and applying,

As some professors say: “It is intuitively obvious to even the most casual observer.”

## References

^{1} Berry, Brett. “Top 10 Secrets of Pascal’s Triangle”. 2018. *Medium*. https://medium.com/i-math/top-10-secrets-of-pascals-triangle-6012ba9c5e23.

^{2} Rout, Siddharth. “Pascal’s Triangle”. 2020. *Future Bound*. https://sidr.hashnode.dev/pascals-triangle.

^{3} user371838, “Pascal’s Triangle’s different usages”. 2017. https://math.stackexchange.com/q/2152960.

^{4} Yadav, Pradeep. “What are some real world examples of the use of the binomial theorem?” 2023. *Quora*. https://qr.ae/pyTVtk.

^{5} Siahaan, Ivander Jeremy. “What’s The Use of Pascal’s Triangle?” 2023. *Medium*. https://medium.com/@ivandessiahaan14/whats-the-use-of-pascal-s-triangle-4532c3687527.

^{6} Llyod, Philip. “How do I solve (√a+√b) ^8 with pascal’s triangle?” 2023. *Quora*. https://qr.ae/pyTJEh.

^{7} Llyod, Philip. “How can you memorize Pascal’s Triangle?” 2023. *Quora*. https://qr.ae/pypNQV.

## Additional Reading

Lazarus. “The Math Behind Galton Board and Pascal’s Triangle”. 2022. *Medium*. https://medium.com/@ozlazarus/the-math-behind-galton-board-and-pascals-triangle-c6f1eca8218b.

The Galton board was invented in 1876 by the Victorian genius Sir Francis Galton and presented an elegant demonstration of how a normal distribution arises from the combination of a large number of random events. It gave Galton key insights into the distribution of human characteristics during his studies on heredity. It also played a pivotal role in his development of statistical theory, concepts that remain fundamental today.

⭐ Liu, Ewen. “Pascal’s Triangle”. 2023. *prezi.com*. https://prezi.com/ns63yt572bto/pascals-triangle/.

## Videos

Pascal’s triangle, which at first may just look like a neatly arranged stack of numbers, is actually a mathematical treasure trove. But what about it has so intrigued mathematicians the world over? Wajdi Mohamed Ratemi shows how Pascal’s triangle is full of patterns and secrets.

Think you know everything about Pascal’s Triangle? Watch this video and be surprised. **You’ll even see how Pi and e are connected!**

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

The featured image on this page is from the MATHS + PHYSICS TUTOR IN HARROGATE AND ONLINE website.