Even More Dice

Carlisle's Casino has a new dice game that everyone is talking about. The player rolls $n$ dice, each with faces numbered $1$ through $m$, and has to hit the sum the dealer picks at random. For every sum the dealer calls, find all the rolls that add up to it.

The first line contains one integer $c$, the number of test cases. Each of the next $c$ lines contains three space separated integers $n$, $m$, and $s$: the number of dice, the number of faces on each die, and the sum that must be rolled.

For each test case, print the line Case x: first, where $x$ is the case number starting at $1$.

Then print every combination of dice values that adds up to $s$, one per line. All $n$ dice must be used. Write a combination with its values comma separated and wrapped in parentheses, arranged from least to greatest when read left to right, and print only one permutation of each combination. For $n = 3$, (1,2,3) is correct while (2,3,1), (2,1,3), (3,1,2), (3,2,1), and (1,3,2) are not.

When more than one combination exists, print them in ascending lexicographic order. The combination with the smaller first value comes first, ties are broken by the second value, and so on. If the sum $s$ cannot be rolled, print only the Case x: line.

• $2 \le n \le 6$
• $3 \le m \le 12$
• $2 \le s \le 80$