Question 1202180
.
The question writes: “There are 8 keys split between two key rings with at least 2 keys on each ring. 
Both key rings can be rotated or turned over. How many different ways can the keys be placed on the key rings?”
How would I solve this?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
If this problem is given to a student, it assumes that the student is just familiar with

    - combinations;
    - circular permutations;
    - and the Fundamental counting principle.


These are pre-requisites.


So, I may assume that you know about the number of permutations  {{{C[n]^k}}} = {{{n!/((n-k)!*k!))}}},

and you know about the number of circular permutations of m objects (m-1)!



Then the solution of the problem is not too hard and you can understand it.


Let k be an integer number between 2 and 6, inclusive:  2 <= k <= 6.


Then you have  {{{C[8]^k}}}  ways to select "k" key on the 1st ring; the rest (8-k) keys
automatically will go to the second ring.


There are (k-1)! circular permutations of  "k"  keys on the 1st ring and 
(8-k-1)!  circular permutations of  (8-k)  keys on the 2nd ring.


For now, it gives  {{{C[8]^k*(k-1)!*(8-k-1)!}}}  possible ways placing the keys.
But my reasoning is not completed, yet.


Turning 1st ring DOES NOT CHANGE the number of placing the keys on the 1st ring;
all possible placing are just accounted for by the formula (k-1)!

Turning 2nd ring DOES NOT CHANGE the number of placing the keys on the 2nd ring;
all possible placing are just accounted for by the formula (8-k-1)!

So, the above formula still remains with no change, even although the turnings are accounted.


Finally, you should make the sum of these addends  {{{C[8]^k*(k-1)!*(8-k-1)!}}}  
over all "k" from 2 to 6, inclusive.


Thus, your final formula is

    number of placing = {{{sum((C[8]^k*(k-1)!*(8-k-1)!),k=2,6)}}}.     (1)


At this point, I completed my explanations. If you want to get the value,
make calculations on your own, following this remarkable formula (1),
which I developed for you with all necessary auxiliary explanations.
</pre>

Solved.


These my reasoning assume that the rings are different (distinguishable),
so for you the pair of rings (1,2) is always different from the pair of rings (2,1).