I guess my approach is same as Limbic~Region as I was using subtraction of lower factorial terms with higher ones.

I copied the input list from one of your previous examples and I have  45700 instead of 4570 but aside from that the implementation should be easy to understand.

I guess if you sort numerator and denominator and subtract them you should be all set. I have not proved this formally but here is a stab at a simple proof that shows sorting and subtracting should work...

Let the fraction be

X! Y! Z!
-------
a! b! c!

WLOG let's assume that X > Y > Z and a > b > c. Also let's assume that
+ b > Y , X > a (weaker assumption: Z > c)...<p>

NOTE: if we divide X!/a! we will have (X-a) elements <p>

To Prove: (X-a) + (Z-c) + (b-Y) is the shortest list one can find 
or in other words 
(X-a) + (Z-c) + (b-Y) <= (X-p) + (Z-q) + (r-Y) for any p,q,r in permut
+ation(a,b,c)

Proof:
From the above equation since b > Y and Z > c, r should be equal to ei
+ther a or b. If r = b then the solution is trivial<p>

If r = a then we get 

(X-a) + (Z-c) + (b-Y) ?<= (X-b) + (Z-c) + (a-Y) 
canceling terms
-a - c + b ?<= -b -c + a
-a + b ?<= -b + a ====> YES
since a > b we see that r = a is not the smallest list so r = b<p>

Similarly we can also show that (X-a) + (Z-c) + (b-Y) <= (X-a) + (Y-c)
+ + (b-Z)
[download]

cheers

SK

PS: I think there will be 47448 elements and not 47444 as you suggested? as you need to count the first element too..