One of the boys had to leave early this morning for a school event, so I was looking for a quick project. With some of the work I did in Mathematica on Taleb’s problem still up on my computer screen, I decided to run through the problem with the boys. The point here wasn’t for them to figure out the solution, but rather to see a neat example of counting techniques used to solve a challenging problem.

I started by explaining the problem and asking them to take a guess at the answer. The boys also had some interesting thoughts about the probability of the balls all ending up in different boxes.

Next we went to Mathematica to walk through my approach to solving the problem. In talking through my approach these ideas from number theory and combinatorics come up:

(1) Partitions of an integer,
(2) Binomial coefficients,
(3) Complimentary counting,
(4) Permutations and combinations, and
(5) Correcting for over counting.

Here’s our quick talk through one solution to Taleb’s problem (and, again, this isn’t intended as a “discovery” exercise, rather we are just walking through my solution) :

To wrap up we returned to the idea of the balls spreading out completely -> a maximum of 1 ball per box. Both boys thought this case was pretty likely and were pretty surprised to find it was less likely than ending up with 3 or more balls in a box!

This problem is little bit on the advanced side for 8th and 6th graders to solve on their own, but they can still understand the ideas in the solution. Also, there are some fun surprises in this problem – the chance of the balls spreading out completely was much lower than they thought, for example – so I think despite being a bit advanced, it is a fun problem to share with kids.