Pascal's Triangle - Numberphile Numberphile English English (auto-generated)

Pascal's Triangle - NumberphileYouTube↗ IA↗(long shot)
Science & Technology 609.2K views 18.8K likes 348 dislikes 10 Mar 2017 Show Contributors 4

Numberphile 4.8M subscribers N/A YouTube↗

Contributed subtitles

English Rishi Kanaujia

Spanish Javier Wagner

Italian Agglomerati di Sodio

Click to Play Video

So today we're going to talk about

so today we're going to talk about

Pascal's triangle. It's one of my

Pascal's triangle it's one of my

favorite things in math - particularly

favorite things in math particularly

because whenever I talk to other

because whenever I talk to other

mathematicians they always have some

mathematicians they always have some

interesting patterns they like talking

interesting patterns they like talking

about. It's almost never the same one.

about it's almost never the same one so

So Pascal's triangle is sort of like an

Pascal's triangle is sort of like an

addition triangle. So we start with

addition triangle so we start with the

the number one and every time I draw one

number 1 and every time I draw one

assume that everything around it is a

assume that everything around it is a

bunch of zeros so what we do is we have

bunch of zeroes so what we do is we have

one and then we add whatever is to the left

one and then we add whatever's to the

of it - so it's neighbor here is 0 and we just

left of it let's neighbor here is zero

drop that down in between - then I do

and we just drop that down in between us

the same thing on the right.

and I do the same thing on the right all

Alright cool now I have two ones in this row

right cool now I have two ones in this

and I'll do the same thing so

row and I'll do the same thing so

there's a 0 we can imagine here so we

there's a zero we can imagine here so we

drop that down.

drop that down there's one in this

There's one in this neighbor now so

neighbor now so we can have a two and

we can have a two and then there's one

then there's one in this zero to its

and the zero to its right we make another one.

right so we make another one and we just

And we just keep doing this for a while and you can see that the triangle just

keep doing this for awhile and you can

keeps getting bigger this is nothing

see that the triangle just keeps getting

that's gonna stop. So we can just keep

bigger this is nothing that's going to

doing this for as long as we want and

stop so you can just keep doing this for

for me as long as I want is when the

as long as we want and for me as long as

first double-digit number shows up

I want is when the first double digit

because that's when adding gets hard.

number shows up because that's when

So Pascal was also a combinatorist so

adding gets hard so Pascal was also a convener tourist so

in particular like problems where "If I have

in particular like problems where I have

N things and I want to choose K of

n things that I want to choose K of them

them, how many different ways can I do

how many different ways can I do that

that?" And it turns out that Pascal's

and it turns out that Pascal's triangle

triangle is really like "childlike adding

it's really like child like adding game

game" as seems to be actually encodes N

as it seems to be actually encodes n

choose K. So if I have N things -

choose K so if I have n things in

in particular let's say i have four things

particular let's say I have four things

and I want to choose let's say one of

and I want to choose let's say one of

them then I go to the fourth row I move

them then I go to the fourth row I move

over one and that's how many ways I can

over one and that's how many ways I can

do it four ways. I if I have four things

do it four ways if I have four things

let's say I have like four circles and

let's say I have like four circles and I

I want to only choose one of them - there are

want to only choose one of them there

four ways i can do that right. I can

are four ways I can do that right I can

choose this one, I can choose that one

choose this one I can choose that one

I can choose that one, I can choose that one

you can choose that one just that one

And it turns out Pascal's triangle says if I go down to the

and it turns out Pascal's triangle says

fourth row and - where it like this is the

if I go down to the fourth row where is

0th row - and I choose one thing so I move

like this is a zero with row and I

one over then that's exactly how many

choose one thing so I move one over and

ways I can do it.

that's exactly how many ways I can do it

(Brady Haran) So if i told you to choose two circles-

so if I told you to choose two circles

Then there are six ways to do it.

then there are six ways to do it you

You notice that the sides are always ones

notice that the sides are always ones

Like, this isn't going to change because

like this isn't going to change because

we just keep doing 1+0 and then oh, look

we just keep doing one plus zero and

1 + 0, 1 plus 0. And this is just gonna keep

then oh look one plus zero 1 plus zero

happening so we can define this entire

it's just going to keep happening so we

diagonal as "+0" because every time I

can define this entire diagonal as plus

go down 1 i'm not changing anything

zero because every time I go down one

But then we get to this row right and we have

I'm not changing anything well then I get to this row right and we have one

one plus one plus one plus one because

plus one plus one plus one because I

I have 0 1 2 3 4 5 so i can define this row

have zero two three four five so I can define this

I felt like it as "+1" then this

row if I felt like it as plus one then

pattern looks less fun right. I have One, a Three

this this pattern looks less fun right I

But the difference between three and one

have one three but the difference

is 2. The difference between six and

between three and one is two the

three is three the difference between

difference between six and three is

10 and 6 is 4 so it turns out this pattern is

three the difference between ten and six is four so it turns out this pattern is

+ 1 + 2 + 3 keeps going. So around here

plus 1 plus 2 Plus 3 keeps going so

One may be tempted to look for more

around here one would be tempted to look

patterns like right here is like plus

for more patterns like right here is

what three plus six but that's when you

like plus what 3 plus 6 but that's when

start to notice what we've been doing

you start to notice what we've been

all along which was just adding the

doing all along which was just adding

number to the top-left like Pascal

contempor to the top left like Pascal

wanted us to anyway. Let's just look at

wanted us to anyway let's just look at

something else one that might be tempted to

something else that one might be tempted

do when they see a bunch of numbers and

to do when they see a bunch of numbers

that's add them in a different way.

and that's add them in a different way

So this is 1. One plus 1 is 2. One plus 2 plus 1 is 4. One plus 3 is 4 plus 3 is 7 plus 1 is 8.

so this is 1 1 plus 1 is 2 1 plus 2 plus 1 is 4 1 plus 3 is 4 plus 3 is 7 plus 1 is 8 you'll notice at some

You'll notice that some point I just stopped adding because we can actually rewrite everything over here as

point I just stopped adding because we can actually rewrite everything over

the powers of 2. So the sum of the rows of

here as the powers of 2 so the sum of

Pascal's triangle is two - like if I'm on

the rows of Pascal's triangle is 2 like

the Nth row and I call the first one row zero then

if I'm on the nth row and I call the first one row zero then the sum of that

the sum of that row is 2 to the N.

row is 2 to the N let's just read these

Let's just read these as numbers let's not add

as numbers let's not add them let's not

them let's not treat them as anything else

treat them as anything else

So the first row is 1 the next row is 11

so the first row is 1 the next row is 11

The next row is a hundred and twenty one. The next row is thirteen thrity-one. Then we get

next row is 121 next row is 1331 and we

Then we get fourteen six-four-one And then we get to this

get 14 6 4 1 and then we get to this so

*laughs* So ,right now these are actually the powers of 11 too

right now these are actually the powers

So this next row looks pretty frightening

of 11 2 so this next row looks pretty

because if I wrote it out just

frightening because if I wrote it out

as it is, I get a pretty big number right

just as it is I get a pretty big number

like once I put in these commas,

right like once I put in these commas

this is around 15 million

this is around 15 million that's not a

(brady) That's not 11 to the 5

little to the bone it's not but it turns

It's not. But it turns out Pascal

out Pascal's you know his triangle

you know his triangle doesn't quit that early

doesn't quit that early so what's

so what's happening here is that

happening here is that when I was

when I was writing these as digits

writing these as digits before I was

before, I was actually considering that

actually considering that place value

place value right - like the ones place

right like the one place the tens place

the tens place the hundreds place

the hundreds place so I'm just going to

So I'm just going to do the same thing where I rewrite these numbers

do the same thing where I rewrite these

as ones and tens and hundreds place

numbers as ones in tens and hundreds

So the first one will be one times 10 to the 0

place so the first one will be 1 times

because this is the 0th row

10 to the 0 because this is the 0th row

Then we have 1 times 10 to the 1 plus 1

then we have 1 times 10 to the 1 plus 1

times 10 to the 0 (1x10^1+1x10^0)

ten to the zero so this one's one this

so this one's 1 this one's 11. Uh, let's do 1 times 10 squared plus 1 times te- oh 2 times 10 to the one my bad - plus 1

one's 11 let's do one times 10 squared plus 1 times 2 times 10 to the 1 my bad

times 10 to 0. That"s a hundred twenty-one

plus 1 times 10 to the 0 that's 121 so

So let's let's skip on down to our friend here

let's let's skip on down to our friend

We do 1 times 10 to the 5 Plus 5 10

here we do 1 times 10 to the 5 plus 5 10

to the 4 plus 10 10 to the 3 plus 10(10) squared? - yeah 5 to the 10 to the 1 and 1 times 10 to the 0

to the 4 plus 10 10 to the 3 plus 10 10 squared yeah 5 to the 10 to the 1 and 1

and this is 11 to the 5

times 10 to 0 and this is 11 to the 5 is

(Brady) So it is 11^5 It is 11^5, it's just hiding

11 to the 5 it is 11 to the 5 it's just

So now I have this really cool thing

hiding so now I have this really cool

that I found extremely recently called

thing that I found extremely recently

um, the hockey sticks method i think

call them the hockey sticks method I

some people call it. It's just a really

think some people call it it's just a

cool trick that you can do on Pascal's

really cool trick that you can do on

triangle another result of the fact that

Pascal's triangle another result of the

it's the coefficients of like a binomial

fact that it's the coefficients of like

expansion which is all of - it will make

a binomial expansion which is all of it

more sense once I draw hockey stick

will make more sense once I draw a

What you do is you pick a one somewhere on the

hockey stick what you do is you pick a

edge and you just go down diagonally so

one somewhere on the edge and you just

i'm going to pick this one and then i'm

go down diagonally so I'm going to pick this one and then I'm going to go down

going to go down diagonally and then I'm just

to a million and then I'm just going to

gonna keep going and then at some point

keep going and then at some point I'm

I'm just gonna veer off in this case I have to

just going to veer off in this case I

veer off because I didn't draw any more

have to veer off because I didn't draw

rows. So the number that I veer off to

any more rows so the number that I veer

is the sum of all the numbers I traced down

off to is the sum of all the numbers I

1 plus 2 Plus 3 plus 4 equals 10 So this row - or this diagonal rather - equals 10

trace down one plus two plus three plus four equals 10 so this row or this diagonal rather equals 10 so I'm gonna

So I'm gonna start with this one

start with this one and it's going to be

and it's gonna be awful real quickly

off real quickly actually I'm just going

actually I'm just gonna all right to

to all right so works on small ones

work on small ones. Uh, let's use this one Because... why not. We can do the other direction, right

let's use this one because why not we can do the other direction right and we

and we get 1 plus two plus three equals 6

get one plus two plus three equals six

I'm gonna do Pascal's triangle mod 2

I'm going to do Pascal's triangle mod

So, brief aside - mod 2 in this case is just

two so brief aside mod two in this case

going to mean that every time a number

is just going to mean that every time a

is odd I write it as a one and every

number is odd I write it as a one and

time a number is even I write it as a zero

every time a number is even I write it

as a zero so cool right that's still a

cool right (Brady) That's still a Pascal Triangle, is that?

tough school song

Yeah, just mod two. It doesn't like

yeah just mod - it doesn't like

preserve the actual value of the numbers

preserved the actual value of the numbers as much

as much, but it totally tells you whether

but it totally tells you whether or not

or not something is even or odd and that's

something's even are out and that's all

all we care about right now.

we care about right now so then I'm

So then I'm going to outline all of the

going to outline all of the diagonals

diagonals that are all ones and all the

that are all ones and all the roads that

rows are all ones So i'm going to do this - which isn't the world's prettiest

are all one so I'm going to do this which isn't the world's prettiest thing

thing but it's pretty nice and this shape should maybe start to

but it's pretty nice and this shape should maybe start to look a little

look a little familiar if you played - I dunno -

familiar if you played on a Legend of

legend of zelda before. It's th-also the Triforce

Zelda before it's also the Triforce but

but more importantly it's sierpinski's triangle

more importantly it's sierpinski

so for those who don't recognize sierpinski's triangle

triangle so for those who don't

I start with an equilateral triangle

recognize your pesky triangle I start with an equilateral triangle I inscribe

I inscribe another one upside down and

another one upside down and then I can

then i can just keep doing this on all

just keep doing this on all of the

of the upward facing triangles as many

upward facing triangles as many times as

times as I want

so yeah that's can just go on for a long time

so yeah this can just go on for a long

(Brady) Cool, yeah? And it turns out that the

time and it turns out that the way that

way that the evens and odds show up

the evens and odds show up in Pascal's

in Pascal's triangle when I convert them

triangle when I convert them to mod - I

to mod 2, I can make Pascal triangle as

can make Pascal's triangle as big as I

big as I want and I'm just going to keep

want and I'm just going to keep getting

getting iterations of sierpinski's triangle

iterations of sequential triangle so

so that's really cool

that's really cool

(Brady) That's awesome! That's really cool by itself but it gets cooler using the mod 2 construction

it's really good by itself but it gets cooler using the mod two construction

There's something else we can do in mod two though

there's something else we can do in mod

Um, in particular we can write down the same

two though in particular we can write down the same things but let's write

things but let's write them now as - we've got ones and zeroes, right?

them now is we've got ones and zeros

And for most people ones and zeros kinda remind them of binary

right and for most people ones and zeros kind of remind them of binary so let's

So, let's just see what happens when we transcribe these from binary

just see what happens when we transcribe

You know one binary - 1 and then 11 in binary is 3 101 is 5

these from binary you know one binary 1 and then 1 1 in binary is 3 1 0 1 is 5 1

111 is 15 and then the next one, 10001 is 17 Now these may not immediately appear pretty cool

1 1 is 15 and then the next one 1 0 0 0 1 is 17 now these may not immediately

but if I kept going the next one is 51 The next one is 85

appear pretty cool but if I kept going the next one is 51 the next one is 85

After that is 255 and here I'm gonna stop and I'm

after that is 255 and here I'm going to

going to look at what these numbers are

stop and I'm going to look at what these

numbers are 3 is prime 5 is prime

Five is prime 15 is 5 times 3 17 is prime 51 is 17 times 3

15 is 5 times 3 17 is prime 51 is 17

85 is 17 x 5 And 255 is 17 x 15

times 3 85 is 17 times 5 and 255 is 17

So you'll notice that every time I get a prime i go back and

times 15 so you'll notice that every time I get a prime I go back and just

just multiply by all the things that came before it

multiply by all the things that came

and I can generate the next you know n minus 1 entries

before and I can generate the next you

So the primes that we have here actually at least the ones

know n minus 1 entries so the prime is that we have here actually at least the

that I'm - that we're certain of - are actually Fermat (pronounced fer-mat) primes

ones that I'm that were certain of are actually format Prime's for Fermont

Or Fermat (pronounced fer-mot) primes rather

Prime's rather which means that they can

Which means that they can be written of the form 2 to the 2 to the n plus 1

be written of the form 2 to the 2 to the

Somehow every time I talk to a mathematician-

n plus 1 somehow every time I talk to a

even 10 minutes ago when i asked a different mathematician for an idea

mathematician even 10 minutes ago when I asked a different mathematician for an

he told me something I've never heard before

idea he told me something I had never

So even if maybe somewhere out in the world all of the things you could

heard before it so even if maybe somewhere out in the world all of the

do with this triangle are done

things you couldn't do with this

I don't know if there's any one person

triangle are done I don't know if

who knows them all and maybe there's

there's any one person who knows them all and maybe there's only one person

only one person who knows one of them

who knows one of them and we just have

and we just have to keep exploring the

to keep exploring this triangle and

triangle and seeing if we can

seeing if we can collaboratively explain

collaboratively explain everything

everything inside of it but it's an

inside of it but it's an infinite triangle right so maybe there are things

infinite triangle right so maybe there

on the hundredth line down that we

are things on the hundredth line down

haven't explored yet

that we haven't explored yet I feel like

I feel like it wouldn't be a numberphile video if somehow

it wouldn't be a numberphile video if

the Fibonacci sequence didn't show up So i'm going to show you where the Fibonacci sequence is in the Pascal's triangle because...

somehow the Fibonacci sequence didn't show up so I'm going to show you what if it been a few sequences in the Pascal

this is a numberphile video

triangle because it's a numberphile

yeah so what we're going to do is we're

video so what we're going to do is we're

gonna add up the shallow diagonals

going to add up the shallow diagonals so

So what that means here is the triangle kind of

what that means here is the triangle

inscribed diagonals already right like I

kind of inscribed diagonals already

could look at this one and just kind of

right like I could look at this one and

follow this but a shallow diagonal means that

just kind of follow this but a shallow

i'm going to kind of go off at a

diagonal means that I'm going to kind of

slightly higher angle that I would have

go off at a slightly higher angle than I

if I was just following the way I drew

would have if I was just following the

the triangle so for example the first

way I drew this triangle so for example

Fibonacci number is one so that's my shallow diagonal

the first Fibonacci number is 1 so that's my shallow diagonal and then this

And then this one doesn't have any other shallow neighbors

one doesn't have any other shallow

so it's just by itself also

neighbors so it's just by itself also

So now we have one,one. So this shallow diagonal is a one and then this one

now we have 1 1 so this shallow diagonal is a 1 and then this one so we do this

So we do this and you get 2

Then we do this one with this two - we get three

then we do this one with this too we get

We do this one with this 3 with this one we get five And you know how this game goes

three we do this one with this three with this one we get five and you know

So we have 1,4 and three here so we're going to get eight

how this game goes so we have one four and three here so we're going to get

We get- I'm goina do one more in case you don't believe me

eight we get I'm going to do one more in case you don't believe me we're going to

We're going to do five plus six plus one and this one down here that I didn't add yet

do five plus six plus one and this one down here that I didn't address and we

And we get 13 (Brady) Cool So there you have it - the two biggest recurrence relations in math that involve addition probably

get 13 cool so they have it the two biggest recurrence relations in math that involve addition probably a

um tied together in the Triangle

tied together in the triangle they're

They're not unique things you know you can have one

not unique things you know you can have

and it encodes the other and it's all

one it encodes the other and it's all

just related in really beautiful ways

just related in really beautiful ways

it's so beautiful you know, this triangle has all

it's so beautiful you know this triangle has all of these

of these really cool properties about it

really cool properties about it and

and again like you said earlier it feels

again like you said earlier it feels

like it's really childlike thing right

like it's really childlike thing right

like I just you know add the things next

like I just you know add the things next

to me and I just plop them down under but

to me and I just popped them down under

I've been playing with this triangle

but I've been playing with this triangle

since... I want to say 8th or 9th grade and

since I want to say eighth or ninth

I learned half of these things in the

grade and I learned half of these things

past month and there are still other

in the past once and there are still

things that I could ramble on about

other things that I could ramble on

about this triangle and this isn't even

about about this triangle and this isn't

my field like I'm still an undergraduate

even my field like I'm still an

I didn't even study this triangle

undergraduate I don't even study this

professionally so just imagine what

triangle professionally so just imagine

people know about the triangle that I

what people know about this triangle

don't even know that they don't even

that I don't even know that they don't

know amongst themselves like it's

even know amongst themselves like it's

Similar videos:

LGR 486 Update! Installing CompactFlash Storage
LGR
580K views 13.3K likes 161 dislikes 22 May 201729% match

Fire Flower Macramé Design 2
Percy palomino tomayquispe
788 views 0 likes 0 dislikes 11 Jun 202029% match

Best Travel Gadgets EVERY Person Should Know!
Natalies Outlet
3.5M views 256K likes 2.7K dislikes 03 Mar 201828% match

🎁 How to keep your 2D ANIMATION ALIVE // Add Life to Your ANIMATION
1on1Animation
690 views 66 likes 0 dislikes 05 Jul 202027% match

Fantastic Quaternions - Numberphile
Numberphile
955.8K views 21.8K likes 297 dislikes 18 Jan 201627% match

❤️LIMPIA CONMIGO DESASTRE COMPLETO DE NOCHE Y PLANES DE DECORACIÓN DE SAN VALENTÍN | Love Meg
TV Art Home Decor
4.3K views 333 likes 4 dislikes 27 Jan 202127% match

Gdańsk Apartment Tour #overseaslife
Ruth Davis
937 views 57 likes 0 dislikes 20 Oct 202027% match

Fire Flower Macramé Design 3
Percy palomino tomayquispe
646 views 0 likes 0 dislikes 06 Jul 202027% match

e to the pi i for dummies
Mathologer
2.5M views 59.3K likes 1.5K dislikes 24 Dec 201527% match

COSAS RARAS E INTERESANTES DE LA TIENDA DE 1$ DOLAR | Claudipia
Claudipia
1.2M views 56.9K likes 904 dislikes 03 Jul 201827% match

MOCKING UP THE NEW FIRST TPI INTAKE THINGS LEARNED
L2theW
2.6K views 68 likes 4 dislikes 05 Jan 202027% match

Trunk Lid Repair
Autorestomod Manic Mechanic Gasoline Media
2.2K views 146 likes 0 dislikes 17 Dec 202027% match

Building a Large Staircase and How to Layout a Stair Stringer
RR Buildings
2M views 27.1K likes -1 dislikes 15 Dec 201927% match

Days Gone - THEY WON'T LET ME LEAVE - Walkthrough Gameplay Part 12
PeteTV
32 views 4 likes 0 dislikes 22 May 201927% match

Macrame Inca Trail 2
Percy palomino tomayquispe
1K views 0 likes 0 dislikes 24 Jul 202027% match

STREET HUSTLE TUTORIAL - 3 Card Monte
Chris Ramsay
584.2K views 15.8K likes 157 dislikes 14 Feb 201727% match

Worm Vernmenting System for Batch Food Processing!
World Composting
1.3K views 84 likes 2 dislikes 19 Aug 202027% match

IBM PC AT: 1988 Upgrade Special! 640K RAM, Adlib etc
LGR
919K views 19.8K likes 258 dislikes 05 May 201727% match

Squaring Primes - Numberphile
Numberphile
1.5M views 37.8K likes -1 dislikes 20 Nov 201827% match

Can You Max Legendary Commanders in Gold Keys 1K Opening | Rise of Kingdoms
Shinchi42
136.7K views 4.7K likes 377 dislikes 30 Jul 202026% match

HOW AÇAI IS MADE IN THE AMAZON (THIS IS CRAZY!) 🇧🇷 BRAZIL | PARÁ STATE
Alexander Travelbum
39K views 5.6K likes 11 dislikes 13 Mar 202126% match

Making a CUSTOM iPhone - in Shenzhen, China
Strange Parts
1.4M views 48.6K likes 0 dislikes 07 Feb 201826% match

🌈 How to make Rainbow Cookies!
iJustine
1.5M views 28.9K likes 511 dislikes 27 Nov 201726% match

Tesla Model 3 Power Frunk Foot Sensor Installation!
Because Tesla
3.2K views 106 likes 3 dislikes 01 May 202026% match

How We Faked a "Boston Dynamics" Robot
Corridor Crew
3.3M views 84.7K likes -1 dislikes 14 Jun 201926% match

RUN BORIS RUN!!!!!!-Boris and the Dark Survival
FergHD
101 views 10 likes 0 dislikes 16 Feb 202026% match

BUILD A DUAL PERMANENT MAGNET ROTOR WIND TURBINE (PART 20)
muddymuddymuddmann
373.1K views 1.7K likes 112 dislikes 21 Aug 201126% match

2000 Round NERF Blaster Mod!
Big Boy Toys
3.7M views 70.6K likes -1 dislikes 26 Jan 201926% match

Alpha 17 - What a Horde Night - 7 Days To Die Alpha 17 EP4 - Pete
PeteTV
183 views 13 likes 1 dislikes 14 Jan 201926% match

QUARANTINE GLOW UP 0-100 *things got crazy lol* | Roxette Arisa
Roxette Arisa
47.7K views 2.4K likes 50 dislikes 03 Apr 202026% match

Settings Privacy About Contact Donate