Math Is Fun Forum

Leonard

Guest

Number of combinations

Im trying to find a formula to calculate the number of spatial arrangements that exists for a molecule consisting of 7 atoms - a central atom and 6 atoms surrounding it. The geometry of the molecule is octahedral. For the 1st case, there exists 6 different types of surrounding atoms : a,b,c,d,e,f. I found all the spatial arrangements , turned out to be 15. This can be calculated via a combination of 2/6 : 6! / 2!(6-2)! = 15

For the 2nd case, we assume the central atom consits of a,a,b,c,d,e. I found the number of isomers to be 9. But how can I calculate this using a formula ? My goal is to find quickly the number of isomers via a formula ( rather than using trial & error ).

Thanks,

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Number of combinations

There is a very nice way to do this, however it can be quite complex.  What mathematics are you in (high school, freshman-sophomore college, upper level college) and what is the name of the math class this is in?  Also, have you ever heard of Polya's Theorem?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Leonard

Guest

Re: Number of combinations

I'm a freshman student. This is actually a chemistry course , Ricky. I am good at algebra and calculus. However, statistics never made much sense to me.

I havent heard of Polya's Theorem, can you please explain how ?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Number of combinations

Unfortunately, not really.  The theorem is in algebraic combinatorics, but it isn't the algebra you're used to.

What this method does is it uses a group and treats all elements of this group as being equivalent.  Now this is extremely nice in your example, because given a geometric object, it isn't very hard to come up with a group representing all the types of rotations and flips (also known as symmetries).  For your object, the octehedron, this group turns out to be extremely nice, the group known as S_4, the group of permutations of order 4.

Now thanks to Polya's Theorem, we first write the characteristic polynomial for this group.  Now it's rather impossible to explain this without you first knowing group theory, and to learn enough to understand this example would take several hours alone.  So without further comment of how it's derived, the characteristic polynomial is:

Now since we weight all orbits the same, we have this is simply:

Plugging in x = 6 (can you figure out what the 7th point is and why we can ignore it?), we get 126 as your answer.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."