Bicycle picture puzzle

Per and Gunnar found an online jigsaw game built on a picture of a bicycle. Both of them are competitive, so they decided to settle who plays it better. The goal of the game is to unscramble the picture.

When a round starts, the picture is cut into $W$ by $H$ rectangles of equal size and scrambled at random. Every scrambled arrangement is equally likely. The player may pick any two rectangles and swap them, and repeats that move until the picture is whole again. The game counts the swaps, and that count is the score for the round.

After finishing a round, Gunnar sent his score to Per together with $W$ and $H$ and told him to beat it. Per soon realized that an unlucky scramble makes Gunnar's score impossible to beat. A lower score is better, so Per wins only if he uses strictly fewer swaps than Gunnar's score.

Assuming Per always plays optimally, find the probability that he beats Gunnar's score.

The first line holds the number of scenarios $T$. Each of the next $T$ lines holds three integers $W$, $H$ and $S$, where $S$ is Gunnar's last score.

• $0 < T \le 150$
• $0 < W \le 5$
• $0 < H \le 4$
• $0 \le S \le W \times H$
• When two scores are compared, the lower one is better.

For each scenario, print on its own line the probability that Per beats Gunnar's score. Write the probability as an irreducible fraction with the numerator and the denominator separated by /. If the answer is an integer, print only the numerator.