Prove this logically

I just now went for my walk and I guess it cleared away a few cobwebs.


When you take away the 4 bowls, imagine that they become "invisible bowls" to be used only when more than one ball is placed in any standard bowl. Now whenever you place an extra ball in a bowl, it resides inside one of the "invisible bowls." Thus you still have 14 slots in which to place the 5 balls in. Also, to illusrate this more clearly, since the bowls are each numbered, let's leave their respective numbers on the "invisible bowls." Now nothing has really changed.


Is it a proof? I doubt it, but it shows how it works.

Any combination of 5 balls in 10 bowls can be represented by a string of 0s and 1s where the 0s represent a bowl and the following 1s represent the number of balls in that bowl. For example 011011010000000 would represent 2 balls in the first bowl, 2 in the second and 1 in the third. The sequence always starts with a 0 and is 15 digits long. The number of ways to form these strings is the same as the number of ways to put 5 1s in 14 positions, which is the number of ways to put 5 balls in 14 bowls.

Except I would have use B's and P's.