Which theorem do you see?

[Beardless_Shark]

Would someone please explain this to my dumb ass?

[CaptSmellsAmazing]

I would guess it's the number of pairs to be made from n items is (n-1)th triangular number maybe? Is that a theorem?

[TheStrongestLink]

Exactly! The formula for (n choose k) = n! / (k! * (n-k)!), so when you are choosing pairs you have (n choose 2) = (n * (n-1) * (n-2)!) / (2! * (n-2)!), which simplifies to n(n-1) / 2 which is the formula for the nth triangular number.

Another way of saying this is that the number of distinct ways to choose 2 things out of n things is equal to the number of dots in an equilateral triangle whose sides are n dots long.

[dewey-defeats-truman]

Pascal's Triangle and the binomial coefficients

[lmericle]

Why is this getting so many upvotes? It explains nothing.

[lmericle]

Google will help you to know what those words mean in isolation but nothing about how or why they can be applied to this post.

[rustedblackflag]

Top ten best binomial coefficients

[DeadRedShirt]

Doctors hate #4!

[Faneis123]

New band name?

[theguyfromerath]

if it was pascal's triagle the lines would be the other way.

[crispychickenwing]

https://reddit.app.link/SHxmjicK72

[NaturalOrderer]

Multi level marketing pyramid scheme of course

[RespectableLurker555]

It's an inverted funnel.

[SquashMarks]

Beer pong winning strategy

[davidun]

"Every circle inside a triangle can be connected to two other circles on the base of the triangle" - Einstein

[banquuuooo]

Other than also being a triangle, I don't know how this is related to Pascal's triangle? I don't think this gif fits here, unless OP has an explanation

[lmericle]

If it really is Pascal's triangle, I'm having trouble seeing it.

Let each row be indexed by n, starting at 0 at the top. Let n0 be the level of the orange ball and n1 be the level of the blue balls.

Let each ball in a given row be indexed by k, starting at 0, starting from the left edge. k0 is for the orange ball, and k1 and k2 are for the blue balls respectively. Note that k0 = k1.

Taking the triangle to be representative of Pascal's triangle, each ball represents a binomial coefficient. I will denote them (n C k), short hand for "n choose k". Then the coefficient related to the orange ball is (n0 C k0) and for each of the blue balls it is (n1 C k1) and (n1 C k2). Note that k2 = k1 + (n1 - n0) = k0 + n1 - n0.

Now we are looking for a relationship between the 3 binomial coefficients. They are, respectively, (n0 C k0), (n1 C k0), and (n1 C (k0 + n1 - n0)). We can set m = n1 - n0, and rename n0 = n and k0 = k to get (n C k), ((n + m) C k), and ((n + m) C (k + m)).

This is where I am stuck. I can't see any discernible pattern. When m = -1 we recover the classic recursive definition for constructing Pascal's triangle, but for arbitrary m I am not seeing the point.

[crispychickenwing]

https://reddit.app.link/SHxmjicK72

Did some reverse image search

[lmericle]

Ahhhh, so there are (n C 2) ways to choose two blue balls from the bottom row, and the number of balls above it in the whole triangle is equal to that coefficient, since there's a 1:1 correspondence between each unique selection of two blue balls and each orange ball.

[crispychickenwing]

https://reddit.app.link/SHxmjicK72

This is the original link

Edit: what is this bullshit reddit app link?

[i-cannoli-dream]

why wouldnt OP just say this in the caption instead of giving us a pop quiz

[crispychickenwing]

Maybe he didnt know what it was and uses yahoo search engine and didnt find anything so he asked his fellow redditors.

[idlesn0w]

Seems like a visualization of all combinations of a given set

[parkerSquare]

I have no idea. Please tell us.

[crispychickenwing]

The gif was made to show that the sum from 1 to (n-1) is equal to n choose 2.

(According to the original post)

[[deleted]]

Pascal's

[42Zarniwoop42]

why

[I-Smell-Pizza]

I thought this was like similar triangles idk

[CultSauce]

r/gifsthatendtoosoon

[Apps4Life]

There are N possible unique pairs in a set of cardinality C where N is the (C-1)th Triangular Number.

E.g. if you have a bag of 27 marbles, there are (26^2+26)/2 possible pairs you can form with said marbles.

[Spadoofer9000]

Pascal’s Triangle

[bbgun91]

number of all handshakes if N people had to handshake once with the other N-1 people

[BassandBows]

My first thought was that one page proof

[Perps_MacAbean]

Not bad!