The Ionian Commission on Procuring Cash has come up with a new gambling game to raise funds for the government. The game is played as follows: Each week, the government will televise a set of $$$m$$$ balls (numbered $$$1 \ldots m$$$) being selected one at a time without replacement. Anyone who wants to play will have to buy a game card. Each card contains $$$n$$$ squares (where $$$n \leq m/2$$$) and in each square are two numbers between 1 and $$$m$$$. No number appears more than once on a card. A sample card is shown in Figure E.1.

Figure E.1: Sample game card with $$$m=10$$$, $$$n=4$$$ and $$$p=5$$$.

After each ball is selected, players cover any square which contains that number (there will be at most one such square on any card). Each game card also has a number $$$p$$$ printed on it, and a contestant wins if he or she covers all $$$n$$$ squares after exactly $$$p$$$ ball selections (i.e., prior to the $$$p^{\text{th}}$$$ selection, they only had $$$n-1$$$ squares covered). Before issuing cards to its citizens, the government would like to get an idea of the likelihood of winning for various values of $$$m, n$$$ and $$$p$$$ so they can set up the payoffs appropriately. They have procured you to write a program to solve this problem.