2019 Gauss 8 Problem 24

There are many ways in which the list 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can be separated into groups. For example, this list could be separated into the four groups 0, 3, 4, 8 and 1, 2, 7 and 6 and 5, 9. The sum of the numbers in each of these four groups is 15, 10, 6, and 14, respectively. In how many ways can the list 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 be separated into at least two groups so that the sum of the numbers in each group is the same?

\textbf{(A)}\ 26\quad \textbf{(B)}\ 29\quad \textbf{(C)}\ 24\quad \textbf{(D)}\ 27\quad \textbf{(E)}\ 32

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.