Pascal's Triangle For Dummies

Blaise Pascal (1622-1662)
Blaise Pascal (1622-1662) | Source

Pascal's triangle is a triangle array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal, but has been known from centuries. It was already known in India, China and Greece.

For the western world is known as Pascal's triangle since 1730. Just take a look how the name array changes according to the country or continent:

Khayyam triangle in Iran, after Omar Khayyam a poet and mathematician from Persia (1048-1131 A.D.)

Yang Hui's triangle in China, after Yang Hui (1238–1298)

In Italy, it is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolo Fontana Tartaglia (1500–1577)

Binomial Expansion

Given (a+b)n as an algebraic binomial to the nth power, we will have an expansion with some coefficients that can be found on the Pascal's triangle. For years we have been struggling for ways of memorizing this set of numbers in a triangle. After checking the original 'building block', we came out with an idea: Mnemonics and graphics.

Consider the expansion:

(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2


Pascal's triangle determines the coefficients which arise in BINOMIAL EXPANSIONS. For example we can build a triangle from scratch positioning our 'duck' on arrow #3. 'Duckier' will be surrounded by numbers 1.

Many High school students have had problems building this triangle. Our teacher would start with number 1 on top. Arrow 2 would be kind of useless but contained two number 1's as coefficients. Which now make sense to us:

(a+b)1 = 1a+1b

Using that duck in place, we can build Pascal's triangle all the way down with a solid foundation.

Bellow we have binomials to the power of 3, 4, 5,6 and 7. Notice the coefficients that follow the Pascal' triangle array with no problem.

Now that you are able to take this graphic-algebraic expansion, we see that the sequence of coefficients on the "eighth arrow" will be like this:
1, 7, 21, 35, 21, 7, 1

How would you calculate (0.999)5 ?

In the next video, a teacher will talk about Pascal and a practical example... in real time.

Tip spoiler: (0.999)5 = (1- 0.001)5 

Solving 0.0999 to the Fifth Power -- watch her trick

A Pascaline, built in 1652
A Pascaline, built in 1652 | Source

About Blaise Pascal

Blaise Pascal (19 June 1623 – 19 August 1662), was a French mathematician, physicist, inventor, writer and Catholic Philosopher. Blaise's father was a tax collector in Rouen, which made Blaise get used to calculations. Actually, at 19 he invented the mechanical calculator which was know as Pascaline. This way, young Pascal would help his father on the tedious accounting calculations. Unfortunately he died at 39 of age, due to tuberculosis and stomach cancer. His legacy was remarkable to engineering, science and philosophy.

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Comments 13 comments

Patty Kenyon profile image

Patty Kenyon 4 years ago from Ledyard, Connecticut

Very Useful information!!! Great Hub!!

tammyswallow profile image

tammyswallow 4 years ago from North Carolina

I could kind of hang in there with the algebra hubs... but.. this is a new beast. What is cool about you is you are a genius and you don't rub it in. Excellent!

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calculus-geometry 4 years ago from Germany

Nice work! Good to meet fellow math nerds on Hubpages!

josh3418 profile image

josh3418 4 years ago from Pennsylvania


You and Janine are busting out those Math hubs huh? :) Very informative and thanks! I was decent at Math in high school, but that was quite some time ago. So, I could use the refreshing! :)

Deborah Brooks profile image

Deborah Brooks 4 years ago from Brownsville,TX

Tammy you are so right Lord is a genius.. I mean look at all that algebra.. I have no earthly idea.. Lol.. Joseph you are too awesome my friend

hugs from Tennessee


Janine Huldie profile image

Janine Huldie 4 years ago from New York, New York

Funny Josh, Isaw this topic when I woke up and immediately clicked to read. I am so a math nerd at heart and this one helped to give this math nerd a bit more of an education. This one did not disappoint and very good explanation of Pascal's Triangle. Voted and shared too!!

LaThing profile image

LaThing 4 years ago from From a World Within, USA

This is going back in time for me! Struggling to understand Pascal's brain... lol! Enjoyed reading it, thank God I don't have to study this any more.....

Voted up and interesting!

Lord De Cross profile image

Lord De Cross 4 years ago Author

Patty Kenyon,

Much appreciated for your comment! Your feed back made our day! Thanks!

Tammy, Is not that we are geniuses, we had a wonderful tutor while growing up. An we thank our parents for believing in our potentials. Thanks friend. Alays proud of a mom like you.


Gee! Now a true happy nerd came to see us, and give his support. We are kind of nerds , but with our HP addiction. Thanks for stopping by!

Josh! LOL! Janine got hold of your comment I see. Ha! This hubs are mainly written in a fun way. Kids don't have time for harsh and cold articles. So we try to use a different approach to the matter. Thanks for reading us though!

Debbie Brooks, A pleasure to have you with us, and making us smile. I do not wear those thick glasses though. Just a normal dude with some ideas. Thank you so much!

Janine Huldie,

You must be always busy with your 2 princesses! We write these hubs on the young fellow's POV. We know they are smart, and we try to keep these hubs interesting. Thanks so much for coming around. You have a wonderful day! Watch for Josh! Lol!

Lord De Cross profile image

Lord De Cross 4 years ago Author

Ops! Lathing Popped up! Lol! Yeah! You don't have to study this messy triangle, that is needed for those those middle school teachers ranting. I guess programming just go straingh to the matter. Hope this hub could help some kids out there. Thanks LA! You always smiling as you write!

ChristyWrites profile image

ChristyWrites 4 years ago from British Columbia, Canada

I remember learning about this in high school - I found it interesting then and now too. I did not know about the different names like you explain in the first part of the hub. Vote up, interesting, and sharing too :)

Lord De Cross profile image

Lord De Cross 4 years ago Author

Thanks Christy writes. We appreciate your feedback. Wonderful to have you with us here. Hop you don't have to see this traingle arrange anymore!

mary615 profile image

mary615 4 years ago from Florida

Now, just let me say this up front.....I love you, I love your writings, but after reading the very first sentence on this article, I knew I was in way over my head!

I will just have to leave this one for the more educated people to comment on, because it is like Greek to me.

I can't help but wonder what your I.Q. is Lord. It has to be HIGH.

I'm looking forward to a nice sweet juicy poem about love next.

Lord De Cross profile image

Lord De Cross 4 years ago Author

Hahaha OKAY Mary! Working on that poem! Don't worry about math, I did mention your name on my last poem," 5 more minutes" Lol! Take care Mary! You are waay to sweet!

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