Glass Balls

Strings of glass balls, as shown in the illustration, are to be fired at. 
What is the total number of different ways in which these sixteen balls may be broken, if we must always break the lowest ball that remains on any string?

Thus, one way would be to break all the four balls on each string in succession, taking the strings from left to right. 
Another would be to break all the fourth balls on the four strings first, then break the three remaining on the first string, then take the balls on the three other strings alternately from right to left, and so on. 
There is such a vast number of different ways (since every little variation of order makes a different way) that one is apt to be at first impressed by the great difficulty of the problem. 
Yet it is really quite simple when once you have hit on the proper method of attacking it. 
How many different ways are there?

See answer

Math Genius