The Hidden Power in Pascal's Triangle

What makes Pascal's triangle so powerful? It has deep connections to the Binomial Theorem and the Central Limit Theorem. And hidden within it are the powers of 2, the Fibonacci sequence, and the fractal Sierpinski's Triangle! Let's explore these patterns and see why they show up in Pascal's Triangle.
00:00 Introduction
00:14 What is Pascal's Triangle?
01:07 Connections to Algebra
04:07 Connections to Probability
06:52 Powers of 2
07:26 Fibonacci Sequence
09:25 Fractal -- Sierpinski's Triangle

[@steves3948]

You’re a great teacher.

[@DrSeanGroathouse]

Thanks, I appreciate that!

[@oafkad]

Another great video. These are always so educational and zen. I love the lack of wild jump cuts or other things. You respect the viewer, much appreciated.

[@atrus3823]

It arises pretty naturally. I discovered it before hearing about it in precalc, when learning about expanding powers of binomials. I worked the first several powers out by hand and looked for patterns. The descending/ascending powers thing leaps right out, but took a bit of staring to notice how the coefficients of lower powers add to the coefficients of higher powers. I pointed it out to my teacher and they were like, “that’s Pascal’s triangles!”

[@smabedi]

This is awesome, Great job!

[@DrSeanGroathouse]

Glad you liked it!

[@Ensrick]

Man, this is blowing my mind. I started out searching for truth in philosophy, and that led to skepticism, empiricism and science. Now science has led me to math, and I'm now convinced math is the most important discipline mankind will ever have. It's the core language of reality, every scientific discipline and all engineering and technological advancements.

[@lovishnahar1807]

ty dr Sean you even helped me by answering my mail, i have suggestion that you should make linear algebra videos why determinant proprties works , why the area exists as such so called determinant operator , you know i mean like its kind way off for newcomer

[@DrSeanGroathouse]

Thanks for the idea, I added it to my list!

[@Phobero]

Ah, Tartaglia's triangle 👍😁

[@user-Moela]

symbol of hyrule

[@compositeboson123]

Dr. Sean, could you explain calculus like you are explaining it to a middle schooler pls?

[@DrSeanGroathouse]

Thanks for the idea, I added it to my list!

[@MostInterestingChannel]

can you teach real analysis to a middle schooler?

[@gqx87]

It's pronouncing shi-erpiñski ,like Fibonacci is pronouncing fee-baw-nachee not fibounaksi

[@donpedro00769]

Fee-baw-nachos*

[@gqx87]

and Einstein is I-n-sh-tine

[@S-payanage]

🦦