Dice Numbers

I have a set of four dice, not marked with spots in the ordinary way, but with Arabic figures, as shown in the illustration. 
Each die, of course, bears the numbers 1 to 6. 
When put together they will form a good many, different numbers. 
As represented they make the number 1246. 

Now, if I make all the different four-figure numbers that are possible with these dice (never putting the same figure more than once in any number), what will they all add up to? 
You are allowed to turn the 6 upside down, so as to represent a 9. I do not ask, or expect, the reader to go to all the labour of writing out the full list of numbers and then adding them up. 
Life is not long enough for such wasted energy. 

Can you get at the answer in any other way?

See answer

Math Genius