# Think outside the box

Discrete Mathematics Level pending

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.

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