# Problem E

Gambling Game

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 1.

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.

## Input

Input consists of a single line containing 3 integers $m, n$ and $p$, as described above, where $2 \leq m \leq 33$, $0 \leq n \leq m/2$ and $0 \leq p \leq m$.

## Output

Output the probability of winning on the $p^{\text {th}}$ selection as a
fraction `x/y` in simplest form.

Sample Input 1 | Sample Output 1 |
---|---|

10 4 5 |
8/45 |

Sample Input 2 | Sample Output 2 |
---|---|

10 4 3 |
0/1 |