Snatoms and Possible Combinations

I recently came across Snatoms by Derek Muller (Famous Veritasium guy).

It made me think about this question:

Let's say we have a 'C' shape which can have 4 attachments to it

'O' shape that can have 2 attachments to it

and 'H' shape that can have only 1 attachment

How many possible valid combinations can be made out of 6 'C', 6 'O' shapes and 12 'H' shapes? Is there any way to generalize this? (Don't think about chemistry concepts while solving this problem, and by "valid" combinations I mean to exclude combinations that are not possible for instance CH5 to CH12)

#Algebra #Geometry #Combinatorics

Easy Math Editor

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Comments

Oh yes! That starts to delve into graph theory. You are basically asking for the number of graphs (up to isomorphism) where the vertices of degree 4, 2, 1 occur at most 6, 6, 12 times.

There are 2 ways to interpret the question
- We care about the positioning of the graph. E.g. $O = O, O = O$ is considered different from $O - O - O - O$ (with the last linked back to the first)
- We only care about the vertices that are used.

Calvin Lin Staff - 5 years, 6 months ago

Hey Calvin, thanks for stopping by. I was travelling so could not reply in time. When I posed the question I had only vertices in my mind :)

Vikram Pandya - 5 years, 6 months ago

Can you take a stab at generalization of this number theory problem over here

Vikram Pandya - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$