Problem 866

[ergias]

To handle what the father does with the pieces, I tried to manually run the case N=4.
However, I could not retrieve the number 994 in the end, and I found a different result depending on the order I placed the pieces.
For instance, if I put 1 then 2 then 3 then 4, the segments' lengths are 1, 2, 3, and 4, so the corresponding result would be $H_1\times H_2\times H_3\times H_4=1\times6\times15\times28 = 2520$.
On the other hand, if I put 2, then 4, then 1, then 3, I create segments of length 1, 1, 2, and 4 so the resulting number would be $1\times1\times6\times28=168$.

How am I supposed to get the expected value 994 ?
Is there something I missed ?

[DJohn]

You are probably missing the meaning of expected. In mathematics the expected number is the average over all possible ways of obtaining the number.

For N=4, there are 4! = 24 orders in which the pieces can be picked up. You've worked out the products for two of those orders. If you find the product for all of the orders, add them together, then divide by 24 (the number of orders), you will have the expected value of the product. And that is indeed 994.

[ergias]

oh ! I indeed misunderstood the word "expected". It is clearer for me now, thanks !