
Answer :
There are, in all,
sixteen balls to be broken, or sixteen
places in the order of breaking.
Call the four strings A, B, C, and
D—order is here of no importance.
The breaking of the balls
on A may occupy any 4 out of these 16 places—that is, the
combinations of 16 things, taken 4 together, will be
13 × 14 × 15 × 16 
= 1,820 
1 × 2 × 3 × 4 
ways for A. In
every one of these cases B may
occupy any 4 out of the remaining 12 places, making
9 × 10 × 11 × 12 
= 495 
1 × 2 × 3 × 4 
ways. Thus
1,820 × 495 = 900,900
different placings are open to A and B. But for every one of these
cases C may occupy
5 × 6 × 7 × 8 
= 70 
1 × 2 × 3 × 4 
different places;
so that
900,900 × 70 = 63,063,000
different placings are open to A, B, and C.
In every one of these
cases, D has no choice but to take the four places that
remain.
Therefore the correct answer is that the balls may be broken in 63,063,000
different ways under the conditions.
