Think outside the box
This week, my math teacher is doing something interesting. Each day, he writes a number \(n\) on the board. He then asks us to write down all ordered lists of \(n\) positive integers such that the sum of the \(n\) numbers is \(2n\). After that, he wants us to go to each list, record the product of the \(n\) integers in that list, and then sum the products over all the lists. If he writes the number 1 on Monday, the number 2 on Tuesday, ..., and the number 5 on Friday, what will be the sum of our answers each day over the whole week?
As an explicit example, on Tuesday, our answer for that day will be \(1 \times 3+2 \times 2+3 \times 1=10\).
This problem was adopted from a similar problem given to me by a friend.