multiplicative principle

[fickle drift]

When trying to find the cardinality of the set $\{(a,b,c): a,b,c\text{ are disjoint 2-element subsets of }{1,2,3,4,5,6}}$ I got $\binom{6}{2}\binom{4}{2}\binom{2}{2}$ but I don't really understand why it works. The multiplicative principle says that $\abs{A\times B\times C}=\abs{A}\abs{B}\abs{C}

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[magic kiln]

Django
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[fickle drift]

But what are sets A, B, and C here?

Normally A is the set of possible values of a, B is the set of possible values of b and likewise for C

But here B differs depending on A

So it's confusing

[fickle drift]

@upper matrix I do t get

[upper matrix]

Well you can chose 2 from 6 for the first two subsets, then you’ll have 4 left, so you can chose 2 from 4, and the remaining 2 chose 2 subset

well

The idea is that no matter what a you choose you can always choose some b

But I guess it’s double counting

Wait no it’s not

Because order matters

That’s why you can just multiply them

A,B,C are just sets of two elements

So to visualize this

For $A$, we can choose 2 distinct elements which there are 6 choose 2 ways to do

[magic kiln]

Magma (N, ^)#Magma4Manager

[upper matrix]

For $B$ we can choose from the 4 that remain, that’s 4 choose 2

[magic kiln]

Magma (N, ^)#Magma4Manager

[upper matrix]

And for $C$, well, we’re left with 1 possible way no matter what

[magic kiln]

Magma (N, ^)#Magma4Manager

[upper matrix]

The fundamental idea is the number of ways we can choose B or C is independent of A

[fickle drift]

So we need a multiplicative principle update

[upper matrix]

While what we can choose from those is dependent on A, the count isn’t

[fickle drift]

Yeah

[upper matrix]

Which is why we’re allowed to multiply

[fickle drift]

it's also 6!/2^3

Take a permutation of 1,2,3,4,5,6 (a1,a2,a3,a4,a5,a6) make a={a1,a2}, b={a3,a4}, c={a5,a6}

And u can probably figure out why we need to divide by 2^3

[upper matrix]

yes

[fickle drift]

And $\frac{(2n)!}{2^n}=\prod_{i=0}^{n-1}\binom{2n-2i}{2}$

[magic kiln]

Django

[upper matrix]

Ah

Django does it make sense now @fickle drift